2016年9月25日日曜日

160925(2)

Ruby


倍数判定法(2)

何倍したものを足すもしくは引けばいいか出力してみた。
素数が大きくなるにつれて、絶対値は大きくなるようだ。

# -*- coding: cp932 -*-

require 'prime'
require 'OpenSSL'

n = 1000
p Prime.each(n).select{|pr| 10 % pr != 0}.map{|pr| OpenSSL::BN.new("10").mod_inverse(pr).to_i}
p 'これだと使いにくいので、下を使用'
ary = []
Prime.each(n).select{|pr| 10 % pr != 0}.each{|pr|
  cp = OpenSSL::BN.new("10").mod_inverse(pr).to_i
  cp -= pr if cp > pr / 2
  ary << cp
}
p ary

出力結果
[1, 5, 10, 4, 12, 2, 7, 3, 28, 26, 37, 13, 33, 16, 6, 55, 47, 64, 22, 8, 25, 9, 68, 91, 31, 75, 11, 34, 89, 118, 96, 14, 15, 136, 110, 49, 117, 52, 18, 163, 172, 58, 138, 20, 190, 67, 159, 23, 70, 24, 217, 226, 180, 79, 27, 244, 194, 253, 85, 88, 215, 280, 94, 222, 298, 236, 243, 35, 106, 36, 257, 112, 38, 115, 39, 278, 361, 41, 42, 379, 388, 130, 44, 133, 45, 320, 415, 139, 327, 48, 341, 442, 50, 151, 51, 469, 157, 487, 383, 390, 169, 57, 514, 404, 411, 178, 60, 541, 425, 184, 432, 62, 568, 577, 193, 453, 196, 66, 595, 202, 474, 205, 622, 631, 71, 72, 509, 220, 74, 223, 676, 530, 685, 77, 232, 551, 558, 81, 730, 739, 247, 579, 83, 84, 256, 600, 86, 259, 614, 793, 265, 621, 635, 820, 92, 93, 656, 847, 663, 286, 677, 874, 684, 295, 892, 698]
"これだと使いにくいので、下を使用"
[1, -2, -1, 4, -5, 2, 7, 3, -3, -11, -4, 13, -14, 16, 6, -6, -20, -7, 22, 8, 25, 9, -29, -10, 31, -32, 11, 34, -38, -13, -41, 14, 15, -15, -47, 49, -50, 52, 18, -18, -19, 58, -59, 20, -21, 67, -68, 23, 70, 24, -24, -25, -77, 79, 27, -27, -83, -28, 85, 88, -92, -31, 94, -95, -33, -101, -104, 35, 106, 36, -110, 112, 38, 115, 39, -119, -40, 41, 42, -42, -43, 130, 44, 133, 45, -137, -46, 139, -140, 48, -146, -49, 50, 151, 51, -52, 157, -54, -164, -167, 169, 57, -57, -173, -176, 178, 60, -60, -182, 184, -185, 62, -63, -64, 193, -194, 196, 66, -66, 202, -203, 205, -69, -70, 71, 72, -218, 220, 74, 223, -75, -227, -76, 77, 232, -236, -239, 81, -81, -82, 247, -248, 83, 84, 256, -257, 86, 259, -263, -88, 265, -266, -272, -91, 92, 93, -281, -94, -284, 286, -290, -97, -293, 295, -99, -299]

160925

Ruby


倍数判定法(1)

tsujimotter さんの記事(http://tsujimotter.hatenablog.com/entry/multiple-of-29)を見て、
倍数判定法が正しいことを確認してみた。

以下で何をしているかは元の記事を読んでほしいのだが、
例えば、19 の倍数判定法は次のことを行う。
n = 2014 のとき、
2014 → 201 + 4 × 2 = 209 → 20 + 9 × 2 = 38
38 は19 の倍数なので、2014 は19 の倍数。
n = 2016 のとき、
2016 → 201 + 6 × 2 = 213 → 21 + 3 × 2 = 27
27 は19 の倍数ではないので、2016 は19 の倍数ではない。

# -*- coding: cp932 -*-

require 'prime'
require 'OpenSSL'

# 1の桁をcp倍して足すことを繰り返す
def f(pr, cp, n)
  m = n / 10 + (n % 10) * cp
  return n % pr if m >= n
  f(pr, cp, m)
end

n = 100
p Prime.each(n).select{|pr| 10 % pr != 0}.map{|pr| OpenSSL::BN.new("10").mod_inverse(pr).to_i}
p 'これだと使いにくいので、下を使用'
ary = []
Prime.each(n).select{|pr| 10 % pr != 0}.each{|pr|
  cp = OpenSSL::BN.new("10").mod_inverse(pr).to_i
  cp -= pr if cp > pr / 2
  ary << cp
}
p ary
p ''

Prime.each(n).select{|pr| 10 % pr != 0}.each{|pr|
  cp = OpenSSL::BN.new("10").mod_inverse(pr).to_i
  cp -= pr if cp > pr / 2
  p [pr, cp, (1..n).map{|i| [i, f(pr, cp, i)]}]
}

出力結果
[1, 5, 10, 4, 12, 2, 7, 3, 28, 26, 37, 13, 33, 16, 6, 55, 47, 64, 22, 8, 25, 9, 68]
"これだと使いにくいので、下を使用"
[1, -2, -1, 4, -5, 2, 7, 3, -3, -11, -4, 13, -14, 16, 6, -6, -20, -7, 22, 8, 25, 9, -29]
""
[3, 1, [[1, 1], [2, 2], [3, 0], [4, 1], [5, 2], [6, 0], [7, 1], [8, 2], [9, 0], [10, 1], [11, 2], [12, 0], [13, 1], [14, 2], [15, 0], [16, 1], [17, 2], [18, 0], [19, 1], [20, 2], [21, 0], [22, 1], [23, 2], [24, 0], [25, 1], [26, 2], [27, 0], [28, 1], [29, 2], [30, 0], [31, 1], [32, 2], [33, 0], [34, 1], [35, 2], [36, 0], [37, 1], [38, 2], [39, 0], [40, 1], [41, 2], [42, 0], [43, 1], [44, 2], [45, 0], [46, 1], [47, 2], [48, 0], [49, 1], [50, 2], [51, 0], [52, 1], [53, 2], [54, 0], [55, 1], [56, 2], [57, 0], [58, 1], [59, 2], [60, 0], [61, 1], [62, 2], [63, 0], [64, 1], [65, 2], [66, 0], [67, 1], [68, 2], [69, 0], [70, 1], [71, 2], [72, 0], [73, 1], [74, 2], [75, 0], [76, 1], [77, 2], [78, 0], [79, 1], [80, 2], [81, 0], [82, 1], [83, 2], [84, 0], [85, 1], [86, 2], [87, 0], [88, 1], [89, 2], [90, 0], [91, 1], [92, 2], [93, 0], [94, 1], [95, 2], [96, 0], [97, 1], [98, 2], [99, 0], [100, 1]]]
[7, -2, [[1, 4], [2, 5], [3, 5], [4, 6], [5, 4], [6, 3], [7, 0], [8, 5], [9, 3], [10, 4], [11, 2], [12, 6], [13, 1], [14, 0], [15, 5], [16, 1], [17, 5], [18, 6], [19, 4], [20, 5], [21, 0], [22, 4], [23, 5], [24, 5], [25, 6], [26, 4], [27, 3], [28, 0], [29, 5], [30, 5], [31, 4], [32, 2], [33, 6], [34, 1], [35, 0], [36, 5], [37, 1], [38, 5], [39, 6], [40, 6], [41, 5], [42, 0], [43, 4], [44, 5], [45, 5], [46, 6], [47, 4], [48, 3], [49, 0], [50, 4], [51, 5], [52, 4], [53, 2], [54, 6], [55, 1], [56, 0], [57, 5], [58, 1], [59, 5], [60, 3], [61, 6], [62, 5], [63, 0], [64, 4], [65, 5], [66, 5], [67, 6], [68, 4], [69, 3], [70, 0], [71, 4], [72, 5], [73, 4], [74, 2], [75, 6], [76, 1], [77, 0], [78, 5], [79, 1], [80, 5], [81, 3], [82, 6], [83, 5], [84, 0], [85, 4], [86, 5], [87, 5], [88, 6], [89, 4], [90, 3], [91, 0], [92, 4], [93, 5], [94, 4], [95, 2], [96, 6], [97, 1], [98, 0], [99, 5], [100, 4]]]
[11, -1, [[1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [6, 5], [7, 4], [8, 3], [9, 2], [10, 1], [11, 0], [12, 1], [13, 2], [14, 3], [15, 4], [16, 5], [17, 5], [18, 4], [19, 3], [20, 2], [21, 1], [22, 0], [23, 1], [24, 2], [25, 3], [26, 4], [27, 5], [28, 5], [29, 4], [30, 3], [31, 2], [32, 1], [33, 0], [34, 1], [35, 2], [36, 3], [37, 4], [38, 5], [39, 5], [40, 4], [41, 3], [42, 2], [43, 1], [44, 0], [45, 1], [46, 2], [47, 3], [48, 4], [49, 5], [50, 5], [51, 4], [52, 3], [53, 2], [54, 1], [55, 0], [56, 1], [57, 2], [58, 3], [59, 4], [60, 5], [61, 5], [62, 4], [63, 3], [64, 2], [65, 1], [66, 0], [67, 1], [68, 2], [69, 3], [70, 4], [71, 5], [72, 5], [73, 4], [74, 3], [75, 2], [76, 1], [77, 0], [78, 1], [79, 2], [80, 3], [81, 4], [82, 5], [83, 5], [84, 4], [85, 3], [86, 2], [87, 1], [88, 0], [89, 1], [90, 2], [91, 3], [92, 4], [93, 5], [94, 5], [95, 4], [96, 3], [97, 2], [98, 1], [99, 0], [100, 1]]]
[13, 4, [[1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [6, 6], [7, 7], [8, 8], [9, 9], [10, 1], [11, 5], [12, 9], [13, 0], [14, 1], [15, 2], [16, 3], [17, 4], [18, 5], [19, 6], [20, 2], [21, 6], [22, 1], [23, 1], [24, 5], [25, 1], [26, 0], [27, 1], [28, 2], [29, 3], [30, 3], [31, 7], [32, 5], [33, 2], [34, 6], [35, 1], [36, 1], [37, 7], [38, 1], [39, 0], [40, 4], [41, 8], [42, 9], [43, 3], [44, 2], [45, 5], [46, 2], [47, 5], [48, 1], [49, 4], [50, 5], [51, 9], [52, 0], [53, 4], [54, 6], [55, 1], [56, 3], [57, 2], [58, 7], [59, 8], [60, 6], [61, 1], [62, 1], [63, 5], [64, 1], [65, 0], [66, 3], [67, 6], [68, 1], [69, 9], [70, 7], [71, 5], [72, 2], [73, 6], [74, 1], [75, 1], [76, 7], [77, 1], [78, 0], [79, 3], [80, 8], [81, 9], [82, 3], [83, 2], [84, 5], [85, 2], [86, 5], [87, 1], [88, 4], [89, 2], [90, 9], [91, 0], [92, 4], [93, 6], [94, 1], [95, 3], [96, 2], [97, 7], [98, 8], [99, 5], [100, 1]]]
[17, -5, [[1, 8], [2, 7], [3, 7], [4, 14], [5, 6], [6, 4], [7, 16], [8, 11], [9, 6], [10, 8], [11, 2], [12, 8], [13, 7], [14, 15], [15, 12], [16, 5], [17, 0], [18, 12], [19, 7], [20, 7], [21, 15], [22, 4], [23, 14], [24, 16], [25, 13], [26, 6], [27, 12], [28, 13], [29, 8], [30, 7], [31, 1], [32, 8], [33, 6], [34, 0], [35, 8], [36, 7], [37, 7], [38, 14], [39, 6], [40, 14], [41, 5], [42, 3], [43, 4], [44, 8], [45, 3], [46, 8], [47, 2], [48, 15], [49, 1], [50, 6], [51, 0], [52, 8], [53, 7], [54, 7], [55, 14], [56, 6], [57, 4], [58, 16], [59, 11], [60, 4], [61, 8], [62, 2], [63, 8], [64, 7], [65, 15], [66, 12], [67, 5], [68, 0], [69, 12], [70, 16], [71, 7], [72, 15], [73, 4], [74, 14], [75, 16], [76, 13], [77, 6], [78, 12], [79, 13], [80, 11], [81, 7], [82, 1], [83, 8], [84, 6], [85, 0], [86, 8], [87, 7], [88, 7], [89, 14], [90, 6], [91, 14], [92, 5], [93, 3], [94, 4], [95, 8], [96, 3], [97, 8], [98, 2], [99, 15], [100, 8]]]
[19, 2, [[1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [6, 6], [7, 7], [8, 8], [9, 9], [10, 1], [11, 3], [12, 5], [13, 7], [14, 9], [15, 3], [16, 7], [17, 3], [18, 3], [19, 0], [20, 2], [21, 4], [22, 6], [23, 8], [24, 1], [25, 5], [26, 9], [27, 7], [28, 3], [29, 2], [30, 3], [31, 5], [32, 7], [33, 9], [34, 3], [35, 7], [36, 3], [37, 3], [38, 0], [39, 4], [40, 4], [41, 6], [42, 8], [43, 1], [44, 5], [45, 9], [46, 7], [47, 3], [48, 2], [49, 6], [50, 5], [51, 7], [52, 9], [53, 3], [54, 7], [55, 3], [56, 3], [57, 0], [58, 4], [59, 8], [60, 6], [61, 8], [62, 1], [63, 5], [64, 9], [65, 7], [66, 3], [67, 2], [68, 6], [69, 1], [70, 7], [71, 9], [72, 3], [73, 7], [74, 3], [75, 3], [76, 0], [77, 4], [78, 8], [79, 5], [80, 8], [81, 1], [82, 5], [83, 9], [84, 7], [85, 3], [86, 2], [87, 6], [88, 1], [89, 9], [90, 9], [91, 3], [92, 7], [93, 3], [94, 3], [95, 0], [96, 4], [97, 8], [98, 5], [99, 7], [100, 1]]]
[23, 7, [[1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [6, 6], [7, 7], [8, 8], [9, 9], [10, 1], [11, 8], [12, 12], [13, 13], [14, 14], [15, 15], [16, 16], [17, 17], [18, 18], [19, 19], [20, 2], [21, 9], [22, 16], [23, 0], [24, 1], [25, 2], [26, 3], [27, 4], [28, 5], [29, 6], [30, 3], [31, 1], [32, 17], [33, 1], [34, 1], [35, 12], [36, 13], [37, 14], [38, 15], [39, 16], [40, 4], [41, 8], [42, 18], [43, 2], [44, 17], [45, 16], [46, 0], [47, 1], [48, 2], [49, 3], [50, 5], [51, 12], [52, 19], [53, 3], [54, 1], [55, 4], [56, 1], [57, 1], [58, 12], [59, 13], [60, 6], [61, 13], [62, 2], [63, 4], [64, 1], [65, 8], [66, 2], [67, 4], [68, 2], [69, 0], [70, 7], [71, 14], [72, 9], [73, 5], [74, 12], [75, 18], [76, 3], [77, 1], [78, 4], [79, 7], [80, 8], [81, 15], [82, 16], [83, 6], [84, 13], [85, 2], [86, 5], [87, 1], [88, 1], [89, 14], [90, 9], [91, 16], [92, 0], [93, 3], [94, 14], [95, 17], [96, 12], [97, 12], [98, 8], [99, 9], [100, 1]]]
[29, 3, [[1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [6, 6], [7, 7], [8, 8], [9, 9], [10, 1], [11, 4], [12, 7], [13, 1], [14, 1], [15, 15], [16, 16], [17, 17], [18, 18], [19, 19], [20, 2], [21, 5], [22, 8], [23, 4], [24, 1], [25, 17], [26, 2], [27, 4], [28, 2], [29, 0], [30, 3], [31, 6], [32, 9], [33, 7], [34, 15], [35, 18], [36, 5], [37, 1], [38, 4], [39, 3], [40, 4], [41, 7], [42, 1], [43, 1], [44, 16], [45, 19], [46, 8], [47, 17], [48, 2], [49, 6], [50, 5], [51, 8], [52, 4], [53, 1], [54, 17], [55, 2], [56, 4], [57, 2], [58, 0], [59, 9], [60, 6], [61, 9], [62, 7], [63, 15], [64, 18], [65, 5], [66, 1], [67, 4], [68, 3], [69, 7], [70, 7], [71, 1], [72, 1], [73, 16], [74, 19], [75, 8], [76, 17], [77, 2], [78, 6], [79, 15], [80, 8], [81, 4], [82, 1], [83, 17], [84, 2], [85, 4], [86, 2], [87, 0], [88, 9], [89, 18], [90, 9], [91, 7], [92, 15], [93, 18], [94, 5], [95, 1], [96, 4], [97, 3], [98, 7], [99, 5], [100, 1]]]
[31, -3, [[1, 4], [2, 7], [3, 22], [4, 5], [5, 14], [6, 13], [7, 1], [8, 7], [9, 4], [10, 4], [11, 6], [12, 15], [13, 23], [14, 2], [15, 11], [16, 14], [17, 11], [18, 7], [19, 5], [20, 7], [21, 3], [22, 12], [23, 21], [24, 21], [25, 7], [26, 15], [27, 12], [28, 4], [29, 6], [30, 22], [31, 0], [32, 4], [33, 7], [34, 22], [35, 5], [36, 14], [37, 13], [38, 1], [39, 7], [40, 5], [41, 4], [42, 6], [43, 15], [44, 23], [45, 2], [46, 11], [47, 14], [48, 11], [49, 7], [50, 14], [51, 7], [52, 3], [53, 12], [54, 21], [55, 21], [56, 7], [57, 15], [58, 12], [59, 4], [60, 13], [61, 22], [62, 0], [63, 4], [64, 7], [65, 22], [66, 5], [67, 14], [68, 13], [69, 1], [70, 1], [71, 5], [72, 4], [73, 6], [74, 15], [75, 23], [76, 2], [77, 11], [78, 14], [79, 11], [80, 7], [81, 14], [82, 7], [83, 3], [84, 12], [85, 21], [86, 21], [87, 7], [88, 15], [89, 12], [90, 4], [91, 13], [92, 22], [93, 0], [94, 4], [95, 7], [96, 22], [97, 5], [98, 14], [99, 13], [100, 4]]]
[37, -11, [[1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [6, 8], [7, 34], [8, 23], [9, 12], [10, 1], [11, 27], [12, 9], [13, 13], [14, 14], [15, 15], [16, 9], [17, 35], [18, 24], [19, 13], [20, 2], [21, 21], [22, 17], [23, 8], [24, 18], [25, 25], [26, 26], [27, 36], [28, 25], [29, 14], [30, 3], [31, 31], [32, 18], [33, 7], [34, 7], [35, 17], [36, 27], [37, 0], [38, 26], [39, 15], [40, 4], [41, 4], [42, 5], [43, 8], [44, 34], [45, 6], [46, 16], [47, 26], [48, 27], [49, 16], [50, 5], [51, 14], [52, 15], [53, 9], [54, 35], [55, 24], [56, 5], [57, 15], [58, 25], [59, 17], [60, 8], [61, 18], [62, 25], [63, 26], [64, 36], [65, 25], [66, 14], [67, 4], [68, 14], [69, 18], [70, 34], [71, 7], [72, 17], [73, 27], [74, 0], [75, 26], [76, 15], [77, 4], [78, 3], [79, 13], [80, 23], [81, 33], [82, 6], [83, 16], [84, 26], [85, 27], [86, 16], [87, 5], [88, 31], [89, 2], [90, 12], [91, 22], [92, 32], [93, 5], [94, 15], [95, 25], [96, 17], [97, 6], [98, 32], [99, 21], [100, 1]]]
[41, -4, [[1, 16], [2, 32], [3, 7], [4, 23], [5, 21], [6, 14], [7, 13], [8, 5], [9, 5], [10, 16], [11, 12], [12, 11], [13, 3], [14, 6], [15, 22], [16, 1], [17, 14], [18, 1], [19, 6], [20, 32], [21, 8], [22, 24], [23, 31], [24, 15], [25, 23], [26, 6], [27, 15], [28, 11], [29, 7], [30, 7], [31, 4], [32, 2], [33, 32], [34, 11], [35, 24], [36, 2], [37, 16], [38, 12], [39, 8], [40, 23], [41, 0], [42, 16], [43, 32], [44, 7], [45, 23], [46, 21], [47, 14], [48, 13], [49, 5], [50, 21], [51, 16], [52, 12], [53, 11], [54, 3], [55, 6], [56, 22], [57, 1], [58, 14], [59, 1], [60, 14], [61, 32], [62, 8], [63, 24], [64, 31], [65, 15], [66, 23], [67, 6], [68, 15], [69, 11], [70, 13], [71, 7], [72, 4], [73, 2], [74, 32], [75, 11], [76, 24], [77, 2], [78, 16], [79, 12], [80, 5], [81, 23], [82, 0], [83, 16], [84, 32], [85, 7], [86, 23], [87, 21], [88, 14], [89, 13], [90, 5], [91, 21], [92, 16], [93, 12], [94, 11], [95, 3], [96, 6], [97, 22], [98, 1], [99, 14], [100, 16]]]
[43, 13, [[1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [6, 6], [7, 7], [8, 8], [9, 9], [10, 1], [11, 11], [12, 12], [13, 13], [14, 14], [15, 15], [16, 16], [17, 17], [18, 18], [19, 19], [20, 2], [21, 15], [22, 22], [23, 23], [24, 24], [25, 25], [26, 26], [27, 27], [28, 28], [29, 29], [30, 3], [31, 16], [32, 29], [33, 33], [34, 34], [35, 35], [36, 36], [37, 37], [38, 38], [39, 39], [40, 4], [41, 17], [42, 3], [43, 0], [44, 1], [45, 2], [46, 3], [47, 4], [48, 5], [49, 6], [50, 5], [51, 18], [52, 16], [53, 1], [54, 11], [55, 12], [56, 13], [57, 14], [58, 15], [59, 16], [60, 6], [61, 19], [62, 29], [63, 2], [64, 15], [65, 22], [66, 23], [67, 24], [68, 25], [69, 26], [70, 7], [71, 2], [72, 33], [73, 3], [74, 16], [75, 33], [76, 33], [77, 34], [78, 35], [79, 36], [80, 8], [81, 15], [82, 34], [83, 4], [84, 6], [85, 3], [86, 0], [87, 1], [88, 2], [89, 3], [90, 9], [91, 22], [92, 35], [93, 5], [94, 19], [95, 16], [96, 1], [97, 11], [98, 12], [99, 13], [100, 1]]]
[47, -14, [[1, 8], [2, 11], [3, 24], [4, 22], [5, 24], [6, 33], [7, 43], [8, 17], [9, 15], [10, 8], [11, 41], [12, 16], [13, 1], [14, 18], [15, 25], [16, 34], [17, 44], [18, 3], [19, 16], [20, 11], [21, 27], [22, 35], [23, 7], [24, 4], [25, 26], [26, 2], [27, 45], [28, 31], [29, 17], [30, 24], [31, 13], [32, 35], [33, 8], [34, 37], [35, 27], [36, 6], [37, 46], [38, 32], [39, 18], [40, 22], [41, 37], [42, 7], [43, 9], [44, 23], [45, 28], [46, 14], [47, 0], [48, 33], [49, 16], [50, 24], [51, 21], [52, 4], [53, 8], [54, 9], [55, 17], [56, 15], [57, 33], [58, 34], [59, 2], [60, 33], [61, 18], [62, 26], [63, 34], [64, 44], [65, 5], [66, 16], [67, 16], [68, 35], [69, 21], [70, 43], [71, 32], [72, 12], [73, 16], [74, 45], [75, 36], [76, 17], [77, 5], [78, 36], [79, 22], [80, 17], [81, 37], [82, 27], [83, 6], [84, 46], [85, 22], [86, 18], [87, 4], [88, 37], [89, 23], [90, 15], [91, 7], [92, 28], [93, 18], [94, 0], [95, 8], [96, 16], [97, 5], [98, 32], [99, 24], [100, 8]]]
[53, 16, [[1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [6, 6], [7, 7], [8, 8], [9, 9], [10, 1], [11, 11], [12, 12], [13, 13], [14, 14], [15, 15], [16, 16], [17, 17], [18, 18], [19, 19], [20, 2], [21, 18], [22, 22], [23, 23], [24, 24], [25, 25], [26, 26], [27, 27], [28, 28], [29, 29], [30, 3], [31, 19], [32, 32], [33, 33], [34, 34], [35, 35], [36, 36], [37, 37], [38, 38], [39, 39], [40, 4], [41, 2], [42, 36], [43, 43], [44, 44], [45, 45], [46, 46], [47, 47], [48, 48], [49, 49], [50, 5], [51, 18], [52, 37], [53, 0], [54, 1], [55, 2], [56, 3], [57, 4], [58, 5], [59, 6], [60, 6], [61, 22], [62, 38], [63, 1], [64, 11], [65, 12], [66, 13], [67, 14], [68, 15], [69, 16], [70, 7], [71, 23], [72, 39], [73, 2], [74, 23], [75, 22], [76, 23], [77, 24], [78, 25], [79, 26], [80, 8], [81, 24], [82, 4], [83, 3], [84, 39], [85, 32], [86, 33], [87, 34], [88, 35], [89, 36], [90, 9], [91, 25], [92, 2], [93, 4], [94, 2], [95, 36], [96, 43], [97, 44], [98, 45], [99, 46], [100, 1]]]
[59, 6, [[1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [6, 6], [7, 7], [8, 8], [9, 9], [10, 1], [11, 7], [12, 12], [13, 13], [14, 14], [15, 15], [16, 16], [17, 17], [18, 18], [19, 19], [20, 2], [21, 8], [22, 14], [23, 2], [24, 24], [25, 25], [26, 26], [27, 27], [28, 28], [29, 29], [30, 3], [31, 9], [32, 15], [33, 8], [34, 27], [35, 8], [36, 36], [37, 37], [38, 38], [39, 39], [40, 4], [41, 1], [42, 16], [43, 14], [44, 28], [45, 27], [46, 4], [47, 4], [48, 48], [49, 49], [50, 5], [51, 7], [52, 17], [53, 2], [54, 29], [55, 8], [56, 1], [57, 4], [58, 2], [59, 0], [60, 6], [61, 12], [62, 18], [63, 24], [64, 3], [65, 36], [66, 16], [67, 48], [68, 29], [69, 6], [70, 7], [71, 13], [72, 19], [73, 25], [74, 9], [75, 37], [76, 14], [77, 49], [78, 8], [79, 12], [80, 8], [81, 14], [82, 2], [83, 26], [84, 15], [85, 38], [86, 28], [87, 5], [88, 1], [89, 18], [90, 9], [91, 15], [92, 8], [93, 27], [94, 8], [95, 39], [96, 27], [97, 7], [98, 4], [99, 24], [100, 1]]]
[61, -6, [[1, 15], [2, 11], [3, 43], [4, 22], [5, 31], [6, 25], [7, 8], [8, 13], [9, 7], [10, 15], [11, 3], [12, 5], [13, 41], [14, 16], [15, 32], [16, 26], [17, 2], [18, 14], [19, 8], [20, 11], [21, 24], [22, 51], [23, 34], [24, 1], [25, 33], [26, 21], [27, 21], [28, 15], [29, 7], [30, 43], [31, 14], [32, 52], [33, 7], [34, 4], [35, 34], [36, 15], [37, 22], [38, 16], [39, 1], [40, 22], [41, 12], [42, 17], [43, 23], [44, 41], [45, 34], [46, 7], [47, 23], [48, 17], [49, 11], [50, 31], [51, 6], [52, 42], [53, 17], [54, 42], [55, 15], [56, 3], [57, 24], [58, 14], [59, 12], [60, 25], [61, 0], [62, 15], [63, 11], [64, 43], [65, 22], [66, 31], [67, 25], [68, 8], [69, 13], [70, 8], [71, 15], [72, 3], [73, 5], [74, 41], [75, 16], [76, 32], [77, 26], [78, 2], [79, 14], [80, 13], [81, 11], [82, 24], [83, 51], [84, 34], [85, 1], [86, 33], [87, 21], [88, 21], [89, 15], [90, 7], [91, 43], [92, 14], [93, 52], [94, 7], [95, 4], [96, 34], [97, 15], [98, 22], [99, 16], [100, 15]]]
[67, -20, [[1, 47], [2, 27], [3, 7], [4, 54], [5, 34], [6, 14], [7, 61], [8, 41], [9, 21], [10, 47], [11, 38], [12, 28], [13, 8], [14, 55], [15, 35], [16, 15], [17, 62], [18, 42], [19, 22], [20, 27], [21, 36], [22, 9], [23, 9], [24, 56], [25, 36], [26, 16], [27, 63], [28, 43], [29, 23], [30, 7], [31, 34], [32, 7], [33, 47], [34, 57], [35, 37], [36, 17], [37, 64], [38, 44], [39, 24], [40, 54], [41, 32], [42, 5], [43, 45], [44, 46], [45, 38], [46, 18], [47, 65], [48, 45], [49, 25], [50, 34], [51, 3], [52, 7], [53, 28], [54, 26], [55, 24], [56, 19], [57, 66], [58, 46], [59, 26], [60, 14], [61, 28], [62, 47], [63, 8], [64, 6], [65, 4], [66, 27], [67, 0], [68, 47], [69, 27], [70, 61], [71, 26], [72, 57], [73, 55], [74, 53], [75, 51], [76, 25], [77, 47], [78, 45], [79, 28], [80, 41], [81, 24], [82, 37], [83, 35], [84, 33], [85, 31], [86, 23], [87, 27], [88, 25], [89, 23], [90, 21], [91, 19], [92, 17], [93, 15], [94, 13], [95, 11], [96, 9], [97, 7], [98, 5], [99, 3], [100, 47]]]
[71, -7, [[1, 12], [2, 24], [3, 5], [4, 43], [5, 32], [6, 1], [7, 22], [8, 15], [9, 8], [10, 12], [11, 42], [12, 2], [13, 51], [14, 44], [15, 25], [16, 3], [17, 23], [18, 16], [19, 8], [20, 24], [21, 35], [22, 13], [23, 52], [24, 4], [25, 16], [26, 31], [27, 24], [28, 17], [29, 1], [30, 5], [31, 17], [32, 6], [33, 53], [34, 33], [35, 11], [36, 32], [37, 25], [38, 16], [39, 11], [40, 43], [41, 21], [42, 61], [43, 8], [44, 26], [45, 4], [46, 33], [47, 26], [48, 8], [49, 12], [50, 32], [51, 14], [52, 62], [53, 41], [54, 8], [55, 41], [56, 34], [57, 24], [58, 2], [59, 13], [60, 1], [61, 7], [62, 34], [63, 34], [64, 12], [65, 42], [66, 35], [67, 17], [68, 21], [69, 14], [70, 22], [71, 0], [72, 12], [73, 24], [74, 5], [75, 43], [76, 32], [77, 1], [78, 22], [79, 15], [80, 15], [81, 12], [82, 42], [83, 2], [84, 51], [85, 44], [86, 25], [87, 3], [88, 23], [89, 16], [90, 8], [91, 24], [92, 35], [93, 13], [94, 52], [95, 4], [96, 16], [97, 31], [98, 24], [99, 17], [100, 12]]]
[73, 22, [[1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [6, 6], [7, 7], [8, 8], [9, 9], [10, 1], [11, 11], [12, 12], [13, 13], [14, 14], [15, 15], [16, 16], [17, 17], [18, 18], [19, 19], [20, 2], [21, 21], [22, 22], [23, 23], [24, 24], [25, 25], [26, 26], [27, 27], [28, 28], [29, 29], [30, 3], [31, 25], [32, 32], [33, 33], [34, 34], [35, 35], [36, 36], [37, 37], [38, 38], [39, 39], [40, 4], [41, 26], [42, 42], [43, 43], [44, 44], [45, 45], [46, 46], [47, 47], [48, 48], [49, 49], [50, 5], [51, 27], [52, 49], [53, 53], [54, 54], [55, 55], [56, 56], [57, 57], [58, 58], [59, 59], [60, 6], [61, 28], [62, 5], [63, 63], [64, 64], [65, 65], [66, 66], [67, 67], [68, 68], [69, 69], [70, 7], [71, 29], [72, 27], [73, 0], [74, 1], [75, 2], [76, 3], [77, 4], [78, 5], [79, 6], [80, 8], [81, 3], [82, 49], [83, 1], [84, 11], [85, 12], [86, 13], [87, 14], [88, 15], [89, 16], [90, 9], [91, 25], [92, 53], [93, 2], [94, 21], [95, 22], [96, 23], [97, 24], [98, 25], [99, 26], [100, 1]]]
[79, 8, [[1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [6, 6], [7, 7], [8, 8], [9, 9], [10, 1], [11, 9], [12, 12], [13, 13], [14, 14], [15, 15], [16, 16], [17, 17], [18, 18], [19, 19], [20, 2], [21, 1], [22, 18], [23, 23], [24, 24], [25, 25], [26, 26], [27, 27], [28, 28], [29, 29], [30, 3], [31, 9], [32, 19], [33, 27], [34, 34], [35, 35], [36, 36], [37, 37], [38, 38], [39, 39], [40, 4], [41, 12], [42, 2], [43, 28], [44, 36], [45, 36], [46, 46], [47, 47], [48, 48], [49, 49], [50, 5], [51, 13], [52, 1], [53, 29], [54, 37], [55, 36], [56, 29], [57, 57], [58, 58], [59, 59], [60, 6], [61, 14], [62, 18], [63, 3], [64, 38], [65, 46], [66, 37], [67, 18], [68, 68], [69, 69], [70, 7], [71, 15], [72, 23], [73, 9], [74, 39], [75, 47], [76, 36], [77, 3], [78, 15], [79, 0], [80, 8], [81, 16], [82, 24], [83, 19], [84, 4], [85, 48], [86, 29], [87, 38], [88, 23], [89, 8], [90, 9], [91, 17], [92, 25], [93, 27], [94, 12], [95, 49], [96, 57], [97, 46], [98, 9], [99, 16], [100, 1]]]
[83, 25, [[1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [6, 6], [7, 7], [8, 8], [9, 9], [10, 1], [11, 11], [12, 12], [13, 13], [14, 14], [15, 15], [16, 16], [17, 17], [18, 18], [19, 19], [20, 2], [21, 21], [22, 22], [23, 23], [24, 24], [25, 25], [26, 26], [27, 27], [28, 28], [29, 29], [30, 3], [31, 28], [32, 32], [33, 33], [34, 34], [35, 35], [36, 36], [37, 37], [38, 38], [39, 39], [40, 4], [41, 29], [42, 42], [43, 43], [44, 44], [45, 45], [46, 46], [47, 47], [48, 48], [49, 49], [50, 5], [51, 3], [52, 52], [53, 53], [54, 54], [55, 55], [56, 56], [57, 57], [58, 58], [59, 59], [60, 6], [61, 28], [62, 56], [63, 63], [64, 64], [65, 65], [66, 66], [67, 67], [68, 68], [69, 69], [70, 7], [71, 32], [72, 57], [73, 73], [74, 74], [75, 75], [76, 76], [77, 77], [78, 78], [79, 79], [80, 8], [81, 33], [82, 58], [83, 0], [84, 1], [85, 2], [86, 3], [87, 4], [88, 5], [89, 6], [90, 9], [91, 34], [92, 59], [93, 1], [94, 11], [95, 12], [96, 13], [97, 14], [98, 15], [99, 16], [100, 1]]]
[89, 9, [[1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [6, 6], [7, 7], [8, 8], [9, 9], [10, 1], [11, 1], [12, 12], [13, 13], [14, 14], [15, 15], [16, 16], [17, 17], [18, 18], [19, 19], [20, 2], [21, 1], [22, 2], [23, 23], [24, 24], [25, 25], [26, 26], [27, 27], [28, 28], [29, 29], [30, 3], [31, 12], [32, 1], [33, 3], [34, 34], [35, 35], [36, 36], [37, 37], [38, 38], [39, 39], [40, 4], [41, 13], [42, 2], [43, 12], [44, 4], [45, 45], [46, 46], [47, 47], [48, 48], [49, 49], [50, 5], [51, 14], [52, 23], [53, 1], [54, 13], [55, 5], [56, 56], [57, 57], [58, 58], [59, 59], [60, 6], [61, 15], [62, 24], [63, 3], [64, 2], [65, 14], [66, 6], [67, 67], [68, 68], [69, 69], [70, 7], [71, 16], [72, 25], [73, 34], [74, 12], [75, 23], [76, 15], [77, 7], [78, 78], [79, 79], [80, 8], [81, 17], [82, 26], [83, 35], [84, 4], [85, 1], [86, 24], [87, 16], [88, 8], [89, 0], [90, 9], [91, 18], [92, 27], [93, 36], [94, 45], [95, 13], [96, 3], [97, 25], [98, 17], [99, 9], [100, 1]]]
[97, -29, [[1, 55], [2, 11], [3, 65], [4, 66], [5, 34], [6, 33], [7, 67], [8, 35], [9, 3], [10, 55], [11, 23], [12, 66], [13, 36], [14, 37], [15, 5], [16, 7], [17, 38], [18, 6], [19, 31], [20, 11], [21, 7], [22, 46], [23, 4], [24, 8], [25, 73], [26, 41], [27, 9], [28, 61], [29, 32], [30, 65], [31, 75], [32, 14], [33, 36], [34, 76], [35, 44], [36, 12], [37, 91], [38, 62], [39, 33], [40, 66], [41, 46], [42, 14], [43, 37], [44, 47], [45, 15], [46, 24], [47, 92], [48, 63], [49, 34], [50, 34], [51, 17], [52, 82], [53, 5], [54, 18], [55, 54], [56, 25], [57, 93], [58, 64], [59, 35], [60, 33], [61, 85], [62, 53], [63, 21], [64, 84], [65, 55], [66, 26], [67, 94], [68, 65], [69, 36], [70, 67], [71, 56], [72, 24], [73, 17], [74, 85], [75, 56], [76, 27], [77, 95], [78, 66], [79, 37], [80, 35], [81, 27], [82, 47], [83, 18], [84, 86], [85, 57], [86, 28], [87, 96], [88, 67], [89, 38], [90, 3], [91, 77], [92, 48], [93, 19], [94, 87], [95, 58], [96, 29], [97, 0], [98, 65], [99, 33], [100, 55]]]

2016年9月22日木曜日

160922

Ruby


Frieze Pattern

A007754 にならって、作ってみた。

def A(k, n)
  p [a = Array.new(n + 1, 1), a.size]
  return if k == 0
  p [b = (1..n).to_a, b.size]
  return if k == 1
  2.upto(k){|i|
    c = (0..n - i).map{|j| (b[j] * b[j + 1] - 1) / a[j + 1]}
    a, b = b, c
    p [b, b.size]
  }
end

n = 15
A(4, n)
p ''
A(n, n)

出力結果
[[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], 16]
[[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], 15]
[[1, 5, 11, 19, 29, 41, 55, 71, 89, 109, 131, 155, 181, 209], 14]
[[2, 18, 52, 110, 198, 322, 488, 702, 970, 1298, 1692, 2158, 2702], 13]
[[7, 85, 301, 751, 1555, 2857, 4825, 7651, 11551, 16765, 23557, 32215], 12]
""
[[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], 16]
[[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], 15]
[[1, 5, 11, 19, 29, 41, 55, 71, 89, 109, 131, 155, 181, 209], 14]
[[2, 18, 52, 110, 198, 322, 488, 702, 970, 1298, 1692, 2158, 2702], 13]
[[7, 85, 301, 751, 1555, 2857, 4825, 7651, 11551, 16765, 23557, 32215], 12]
[[33, 492, 2055, 5898, 13797, 28248, 52587, 91110, 149193, 233412, 351663], 11]
[[191, 3359, 16139, 52331, 136415, 307871, 626219, 1176779, 2077151, 3484415], 10]
[[1304, 26380, 143196, 517412, 1486768, 3666204, 8088260, 16383796, 31008072], 9]
[[10241, 234061, 1415821, 5639201, 17704801, 47352781, 112609421, 244580161], 8]
[[90865, 2314230, 15430835, 67153000, 228675645, 659272730, 1681053055], 7]
[[898409, 25222469, 183754199, 867349799, 3183754229, 9841738169], 6]
[[9791634, 300355398, 2373373752, 12075744186, 47527637790], 5]
[[116601199, 3879397705, 33043478329, 180268812991], 4]
[[1506023953, 54011212472, 493278801183], 3]
[[20967734143, 806288789375], 2]
[[313009988192], 1]

2016年9月19日月曜日

160919(2)

新たな整数列(18)

https://oeis.org/A276730
https://oeis.org/A276731
が追加されました。

160919

Ruby


Bernoulli number(2)

次の3 通りで求めてみた。
①定義とBn = 0 (n が3 以上の奇数)から導く方法
②「ベルヌーイ数とゼータ関数」の第2章 2.2 に載っているアルゴリズム
③「ベルヌーイ数とゼータ関数」の命題1.15 を利用する方法

def c(n, r)
  return 1 if r == 0
  return c(n, n - r) if r > n - r
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end

def bernoulli_0(n)
  return 1r / 2 if n == 1
  return 0 if n % 2 == 1
  a = [1r]
  (1..n / 2).each{|i| a << 1r / 2 - (0..i - 1).inject(0){|s, j| s + c(2 * i + 1, 2 * j) * a[j]} / (2 * i + 1)}
  a[-1]
end

def bernoulli_1(n)
  return 1r / 2 if n == 1
  return 0 if n % 2 == 1
  a = []
  (0..n).each{|i|
    a << 1r / (i + 1)
    i.downto(1){|j| a[j - 1] = j * (a[j - 1] - a[j])}
  }
  a[0]
end

def bernoulli_2(n)
  return 1r / 2 if n == 1
  return 0 if n % 2 == 1
  a = [1, 1r / 6]
  (2..n / 2).each{|i| a << - (1..i - 1).inject(0){|s, j| s + c(2 * i, 2 * j) * a[i - j] * a[j]} / (2 * i + 1)}
  a[n / 2]
end

n = 100
p ary0 = (0..n).map{|i| bernoulli_0(i)}
p ary0 == (0..n).map{|i| bernoulli_1(i)}
p ary0 == (0..n).map{|i| bernoulli_2(i)}

出力結果
[(1/1), (1/2), (1/6), 0, (-1/30), 0, (1/42), 0, (-1/30), 0, (5/66), 0, (-691/2730), 0, (7/6), 0, (-3617/510), 0, (43867/798), 0, (-174611/330), 0, (854513/138), 0, (-236364091/2730), 0, (8553103/6), 0, (-23749461029/870), 0, (8615841276005/14322), 0, (-7709321041217/510), 0, (2577687858367/6), 0, (-26315271553053477373/1919190), 0, (2929993913841559/6), 0, (-261082718496449122051/13530), 0, (1520097643918070802691/1806), 0, (-27833269579301024235023/690), 0, (596451111593912163277961/282), 0, (-5609403368997817686249127547/46410), 0, (495057205241079648212477525/66), 0, (-801165718135489957347924991853/1590), 0, (29149963634884862421418123812691/798), 0, (-2479392929313226753685415739663229/870), 0, (84483613348880041862046775994036021/354), 0, (-1215233140483755572040304994079820246041491/56786730), 0, (12300585434086858541953039857403386151/6), 0, (-106783830147866529886385444979142647942017/510), 0, (1472600022126335654051619428551932342241899101/64722), 0, (-78773130858718728141909149208474606244347001/30), 0, (1505381347333367003803076567377857208511438160235/4686), 0, (-5827954961669944110438277244641067365282488301844260429/140100870), 0, (34152417289221168014330073731472635186688307783087/6), 0, (-24655088825935372707687196040585199904365267828865801/30), 0, (414846365575400828295179035549542073492199375372400483487/3318), 0, (-4603784299479457646935574969019046849794257872751288919656867/230010), 0, (1677014149185145836823154509786269900207736027570253414881613/498), 0, (-2024576195935290360231131160111731009989917391198090877281083932477/3404310), 0, (660714619417678653573847847426261496277830686653388931761996983/6), 0, (-1311426488674017507995511424019311843345750275572028644296919890574047/61410), 0, (1179057279021082799884123351249215083775254949669647116231545215727922535/272118), 0, (-1295585948207537527989427828538576749659341483719435143023316326829946247/1410), 0, (1220813806579744469607301679413201203958508415202696621436215105284649447/6), 0, (-211600449597266513097597728109824233673043954389060234150638733420050668349987259/4501770), 0, (67908260672905495624051117546403605607342195728504487509073961249992947058239/6), 0, (-94598037819122125295227433069493721872702841533066936133385696204311395415197247711/33330)]
true
true

2016年9月18日日曜日

160918(4)

整数列のLINKS の編集(84)

https://oeis.org/A271229
https://oeis.org/A271230
のLINKS を編集しました。

160918(3)

Ruby


Bernoulli number(1)

「ベルヌーイ数とゼータ関数」(荒川恒男・伊吹山知義・金子昌信 著)
の第2章 2.2 に載っているアルゴリズムを用いて計算してみた。
(ちなみに、この本ではB1 = 1 / 2)
計算過程も含めて出力してみたので、第2章 2.2 に載っている逆三角形
と見比べてほしい。

def bernoulli(n)
  ary = []
  a = []
  (0..n).each{|i|
    a << 1r / (i + 1)
    i.downto(1){|j| a[j - 1] = j * (a[j - 1] - a[j])}
    p [i, a]
    ary << a[0] # Bn = a[0]
  }
  p 'Bernoulli number'
  ary
end

p bernoulli(40)

出力結果
[0, [(1/1)]]
[1, [(1/2), (1/2)]]
[2, [(1/6), (1/3), (1/3)]]
[3, [(0/1), (1/6), (1/4), (1/4)]]
[4, [(-1/30), (1/30), (3/20), (1/5), (1/5)]]
[5, [(0/1), (-1/30), (1/20), (2/15), (1/6), (1/6)]]
[6, [(1/42), (-1/42), (-3/140), (2/35), (5/42), (1/7), (1/7)]]
[7, [(0/1), (1/42), (-1/28), (-1/105), (5/84), (3/28), (1/8), (1/8)]]
[8, [(-1/30), (1/30), (1/140), (-4/105), (0/1), (5/84), (7/72), (1/9), (1/9)]]
[9, [(0/1), (-1/30), (1/20), (-1/105), (-1/28), (1/140), (7/120), (4/45), (1/10), (1/10)]]
[10, [(5/66), (-5/66), (1/220), (8/165), (-5/231), (-29/924), (49/3960), (28/495), (9/110), (1/11), (1/11)]]
[11, [(0/1), (5/66), (-5/44), (7/165), (5/132), (-9/308), (-7/264), (8/495), (3/55), (5/66), (1/12), (1/12)]]
[12, [(-691/2730), (691/2730), (-1017/20020), (-44/455), (200/3003), (295/12012), (-343/10296), (-28/1287), (27/1430), (15/286), (11/156), (1/13), (1/13)]]
[13, [(0/1), (-691/2730), (691/1820), (-2663/15015), (-629/12012), (1543/20020), (67/5720), (-1576/45045), (-87/5005), (125/6006), (55/1092), (6/91), (1/14), (1/14)]]
[14, [(7/6), (-7/6), (601/1820), (368/1365), (-35/143), (-41/12012), (3997/51480), (4/6435), (-27/770), (-27/2002), (121/5460), (22/455), (13/210), (1/15), (1/15)]]
[15, [(0/1), (7/6), (-7/4), (1247/1365), (15/364), (-1013/4004), (133/3432), (464/6435), (-6/715), (-205/6006), (-11/1092), (3/130), (13/280), (7/120), (1/16), (1/16)]]
[16, [(-3617/510), (3617/510), (-809/340), (-244/255), (5350/4641), (-3515/18564), (-38759/175032), (140/1989), (1539/24310), (-75/4862), (-605/18564), (-11/1547), (169/7140), (91/2040), (15/272), (1/17), (1/17)]]
[17, [(0/1), (-3617/510), (3617/340), (-1511/255), (107/204), (32421/30940), (-9653/26520), (-18536/109395), (1113/12155), (775/14586), (-55/2652), (-95/3094), (-13/2856), (49/2040), (35/816), (8/153), (1/18), (1/18)]]
[18, [(43867/798), (-43867/798), (922191/45220), (43416/11305), (-46705/6783), (3035/1596), (1104901/1511640), (-88508/188955), (-51219/461890), (9597/92378), (10769/251940), (-517/20995), (-3887/135660), (-91/38760), (125/5168), (40/969), (17/342), (1/19), (1/19)]]
[19, [(0/1), (43867/798), (-43867/532), (1623817/33915), (-220565/27132), (-47569/9044), (7179/2584), (441824/1322685), (-74976/146965), (-105155/1939938), (38555/352716), (9643/293930), (-7423/271320), (-343/12920), (-7/15504), (352/14535), (34/855), (9/190), (1/20), (1/20)]]
[20, [(-174611/330), (174611/330), (-6132631/29260), (-276356/21945), (181300/3553), (-5472055/298452), (-563801/255816), (98908/31977), (-170073/3233230), (-326115/646646), (-1331/352716), (3223/29393), (169/7140), (-377/12920), (-885/36176), (8/6783), (289/11970), (51/1330), (19/420), (1/21), (1/21)]]
[21, [(0/1), (-174611/330), (174611/220), (-10405289/21945), (619991/5852), (14842153/497420), (-9937529/426360), (179992/159885), (52449/17765), (-738425/1939938), (-164455/352716), (3567/92378), (21177/198968), (6601/426360), (-5165/170544), (-5024/223839), (34/13167), (351/14630), (57/1540), (10/231), (1/22), (1/22)]]
[22, [(854513/138), (-854513/138), (12988703/5060), (-21376/345), (-46300675/100947), (79819919/403788), (-91434539/29418840), (-84078428/3677355), (3255471/817190), (410187/163438), (-25619891/40562340), (-1381937/3380195), (2494609/34321980), (988897/9806280), (10825/1307504), (-2520/81719), (-12427/605682), (255/67298), (361/15180), (190/5313), (21/506), (1/23), (1/23)]]
[23, [(0/1), (854513/138), (-854513/92), (21491081/3795), (-4486325/3036), (-21955485/134596), (25942189/115368), (-129558928/3677355), (-7542618/408595), (155735/25806), (169939/89148), (-5441531/6760390), (-2132689/6240360), (970543/9806280), (1967/20976), (7456/3677355), (-6698/216315), (-6291/336490), (171/35420), (125/5313), (35/1012), (11/276), (1/24), (1/24)]]
[24, [(-236364091/2730), (236364091/2730), (-1552922421/41860), (32209348/10465), (337992250/69069), (-678649375/276276), (1108107805/4499352), (4638284/24453), (-626165937/10623470), (-25306815/2124694), (8374289/1158924), (120659/96577), (-2836327/3120180), (-242333/891480), (8147/68816), (105272/1225785), (-289/86526), (-1479/48070), (-361/21252), (152/26565), (147/6325), (77/2300), (23/600), (1/25), (1/25)]]
[25, [(0/1), (-236364091/2730), (236364091/1820), (-2523785339/31395), (582059503/25116), (119063489/460460), (-328836059/131560), (11876842408/19684665), (249912591/2187185), (-3196365475/44618574), (-38510285/8112468), (609029/79534), (762863/1248072), (-194383/203320), (-943565/4635696), (1261664/9561123), (218314/2812095), (-4941/624910), (-399/13156), (-1063/69069), (427/65780), (2057/89700), (253/7800), (12/325), (1/26), (1/26)]]
[26, [(8553103/6), (-8553103/6), (1139644561/1820), (-107638072/1365), (-18141585/299), (886838131/25116), (-6661850629/1184040), (-22816268/13455), (274348161/336490), (900387/38038), (-3001800791/40562340), (6694963/3380195), (23377679/3120180), (10369/297160), (-10368135/10816624), (-283208/2028117), (1107737/7873866), (303399/4374370), (-1083/92092), (-684/23023), (-686/49335), (77/10764), (529/23400), (92/2925), (25/702), (1/27), (1/27)]]
[27, [(0/1), (8553103/6), (-8553103/4), (1827648887/1365), (-150546745/364), (184550647/8372), (227017945/7176), (-1501623488/148005), (-7030668/16445), (2264592835/2624622), (-29878651/477204), (-65974293/965770), (693979/90440), (361949/52440), (-163361/356592), (-1485856/1601145), (-146098/1789515), (636417/4374370), (513/8372), (-15/1001), (-381/13156), (-121/9660), (253/32760), (152/6825), (25/819), (13/378), (1/28), (1/28)]]
[28, [(-23749461029/870), (23749461029/870), (-7089687053/580), (842271812/435), (6768026800/7917), (-18511988255/31668), (74587309657/624312), (1136903908/78039), (-11414065671/953810), (172090155/190762), (10692144551/13838916), (-153207835/1153243), (-225197401/3934140), (13574561/1124040), (20818639/3447056), (-927208/1077205), (-90734729/103791870), (-348483/11532430), (842213/5722860), (536104/10015005), (-126343/7153575), (-219439/7803900), (-53429/4750200), (4876/593775), (3125/142506), (325/10962), (27/812), (1/29), (1/29)]]
[29, [(0/1), (-23749461029/870), (23749461029/580), (-11254630547/435), (2924743559/348), (-43597042623/52780), (-20217548561/45240), (71527976584/390195), (-361830837/43355), (-6317754175/572286), (208735615/104052), (1361390201/2306486), (-387567557/2129064), (-1117854983/25852920), (156824675/10341168), (9750176/1938969), (-670106/570285), (-9284517/11532430), (17613/1213940), (293245/2003001), (88151/1907620), (-51667/2601300), (-18469/678600), (-664/65975), (205/23751), (169/7830), (117/4060), (14/435), (1/30), (1/30)]]
[30, [(8615841276005/14322), (-8615841276005/14322), (378639019356093/1384460), (-17380760743952/346115), (-2765846511545/207669), (9194040084569/830676), (-123601607588137/46280520), (-378147458164/5785065), (5661825323859/29568110), (-6040416093/203918), (-130271599469/16128060), (3683074417/1344005), (59711076113/165002460), (-9894184391/47143560), (-3018551695/106858736), (341579704/20036013), (28024619/7071534), (-5532123/3928630), (-3567763/4908540), (297844/5644821), (2125424/14784055), (126973/3225612), (-151823/7012200), (-69092/2629575), (-39625/4417686), (3055/339822), (2673/125860), (126/4495), (29/930), (1/31), (1/31)]]
[31, [(0/1), (8615841276005/14322), (-8615841276005/9548), (596303510772251/1038345), (-160974527939935/830676), (7043839847269/276892), (540201920089/79112), (-147653672046128/40495455), (1756630755402/4499495), (18392665376975/124186062), (-1003250897879/22579284), (-835776444953/206976770), (587853689813/191055480), (21641566527/172859720), (-45389402737/207431664), (-1963487072/143736615), (3481362578/194467185), (125753553/43214930), (-310479/197780), (-1739215/2699697), (1005025/11827244), (2253383/16128060), (15433/467480), (-20236/876525), (-1775/70122), (-1352/169911), (117/12586), (1127/53940), (203/7440), (15/496), (1/32), (1/32)]]
[32, [(-7709321041217/510), (7709321041217/510), (-5645818769488043/811580), (78381346456972/55335), (37011879077950/168113), (-3358366543488455/14121492), (27161485402723/417384), (-3734407504708/1513017), (-510603999545313/152982830), (23291844682185/30596566), (27629378745727/383847828), (-1630585580675/31987319), (341000734763/1623971580), (473383384539/154663960), (-768718294563/8228122672), (-1639873520872/7713865005), (-1017074521/2722540590), (1084420089/60500902), (2650101/1384460), (-578208/346115), (-174391/311025), (27016591/241920900), (31156513/231402600), (785588/28925325), (-168125/6942078), (-6500/267003), (-486/69223), (189/19778), (841/40920), (145/5456), (31/1056), (1/33), (1/33)]]
[33, [(0/1), (-7709321041217/510), (7709321041217/340), (-8834901408462277/608685), (818912043504603/162316), (-18566786190720487/23535820), (-2145268463452009/20173560), (607227019180712/7565085), (-10508379435681/840565), (-178860729240925/91789698), (15956655812845/16689036), (-136536617949/9139234), (-978930707963/19684504), (1872413032063/463991880), (46778413295/16872432), (-1287864273824/4628319003), (-265715469238/1361270295), (57126741291/5142576670), (9368631549/541323860), (27155181/27066193), (-133067499/77331980), (-4594256843/9596195700), (334040339/2503355400), (147827776/1147371225), (3000260/137684547), (-7971899/317733570), (-550251/23535820), (-62083/10086780), (13601/1391280), (1875/92752), (155/5984), (16/561), (1/34), (1/34)]]
[34, [(2577687858367/6), (-2577687858367/6), (67894941959587/340), (-11191744708984/255), (-431245550675215/121737), (164807798297293/28644), (-105779053462256573/60520680), (1084435995174364/7565085), (104816115507537/1681130), (-6532598785551/336226), (-473438541257/83445180), (6652193244301/6953765), (-13974856491401/147633780), (-1790568956813/42181080), (437238485465/61865584), (8898226024/3866599), (-229830918617/544508118), (-51537067719/302504510), (4772961139/231995940), (1317371764/81198579), (11140682/57998985), (-21890429/12654324), (-1001660971/2503355400), (47198852/312919425), (101244125/826107282), (97565/5776974), (-605799/23535820), (-37701/1681130), (-841/157080), (6467/649264), (12493/628320), (496/19635), (33/1190), (1/35), (1/35)]]
[35, [(0/1), (2577687858367/6), (-2577687858367/4), (105697073459989/255), (-30092810459185/204), (4213775170171353/162316), (198527138848087/139128), (-14764512136334176/7565085), (325555556912904/840565), (19482261716135/1008678), (-3917458641503/183396), (26927816029243/13907530), (10209601679537/12837720), (-6573264916547/42181080), (-528420013681/16872432), (1593021999776/173996955), (17697429182/10235115), (-158451923169/302504510), (-4498335063/31842580), (2274260315/81198579), (229268333/15466396), (-10836397/21090540), (-552299/323640), (-11332228/34768825), (4115875/25033554), (526435/4539051), (14625/1176791), (-264271/10086780), (-29899/1391280), (-2995/649264), (1271/125664), (128/6545), (44/1785), (17/630), (1/36), (1/36)]]
[36, [(-26315271553053477373/1919190), (26315271553053477373/1919190), (-8222082433140118821/1279460), (479045504390869788/319865), (130785739894457300/3262623), (-2029755955421007845/13050492), (197903031977441655193/3814472376), (-259418672105285140/43346277), (-18491395258437068817/15363847070), (1601020788482746725/3072769414), (-54955972131102955/1676056044), (-2567138088186001/139671337), (15648566238296693/4512458580), (681411178820827/1289273880), (-765395527824139/3953773232), (-68250051266536/3706662405), (148312964303501/14390571690), (1795365934671/1598952410), (-2071869164443/3534526380), (-97567361624/883631595), (119921527537/3576604075), (51608457901/3901749900), (-126103549/113094200), (-210954436/127230975), (-713703125/2778724494), (18669625/106874019), (9514665/87082534), (104265/12440362), (-682051/25738680), (-70615/3431824), (-18259/4649568), (496/48433), (121/6290), (187/7770), (35/1332), (1/37), (1/37)]]
[37, [(0/1), (-26315271553053477373/1919190), (26315271553053477373/1279460), (-12745379713118458459/959595), (3698785153161779183/767676), (-20087877601953700037/21750820), (-9984854869351353/6214520), (869656204850562154264/16688316645), (-23175912707428338687/1854257405), (11949669049953887225/64528157694), (5895721755116780455/11732392308), (-153444418321389515/1955398718), (-21371906191835281/1804983432), (24462964435947163/5586853480), (1446512606787115/6704224176), (-6012827171665312/28911966759), (-46043735779382/8503519635), (44170101043425/4157276266), (105935738169/198912740), (-26811141956723/43651400793), (-3313591015609/41572762660), (35971260904733/963732225300), (966281025017/83802802200), (-16931868536/10475350275), (-666523495/419014011), (-3923492677/20306063610), (273935727/1504152860), (726935531/7091006340), (4616423/978069840), (-1738475/65204656), (-248465/12620256), (-9088/2760681), (2596/250971), (8381/442890), (595/25308), (18/703), (1/38), (1/38)]]
[38, [(2929993913841559/6), (-2929993913841559/6), (294857320048274438263/1279460), (-53977876347278501536/959595), (7134184511439685/9139), (3578930853369592475/767676), (-95112988481654565863/55930680), (1687213261158668404/6991335), (6260409368416144773/285270370), (-11078927789402666187/741702962), (98487582561597348229/58661961540), (1710432681589587703/4888496795), (-25561802555592199/237497820), (-1530495990317911/429757960), (72475589995478625/15643189744), (-273095609474120/2933098077), (-4813184163035299/23809854978), (2763548745453/426699910), (157221014825487/15316280980), (-255953076/33296263), (-19138831594708/31179571995), (-343378980791/6802815708), (321310631971/8109948600), (308221527908/31426050825), (-412788125/203844654), (-875884345/580173246), (-2642772501/19553987180), (261360351/1396713370), (609248153/6357453960), (109069/77060048), (-21909839/820316640), (-482608/25634895), (-29403/10875410), (20009/1919190), (6125/329004), (210/9139), (37/1482), (1/39), (1/39)]]
[39, [(0/1), (2929993913841559/6), (-2929993913841559/4), (455443615848938396081/959595), (-44757008082536826815/255892), (9151158782827676543/255892), (-284756725155498799/219336), (-10592463472445174416/6991335), (334585688357146134/776815), (-57656278674155113685/2225108886), (-4994755545598634035/404565252), (3912560524435871787/1396713370), (350319504797410317/3008305720), (-150313065679722827/1289273880), (4985930691763/1046064), (189857358823248416/43996471155), (-938971709915578/2588027715), (-2391685981711659/13227697210), (23004363635523/1392389180), (7195134951825/765814049), (-208939239051/437608028), (-3608657740853/6105091020), (-488791159351/20704221720), (2980450385348/73327451925), (4924122475/606847878), (-4771079638/2030606361), (-213488307/150415286), (-692085359/8380280220), (7087339/37287120), (75677265/847660528), (-85343/54687776), (-35968/1349205), (-41932/2330445), (-12427/5757570), (3451/329004), (837/45695), (111/4940), (19/780), (1/40), (1/40)]]
[40, [(-261082718496449122051/13530), (261082718496449122051/13530), (-82622815315007896987/9020), (15700351621967437588/6765), (-9138915566031900914350/86555469), (-53245099482846805949215/346221876), (18219172354548840589327/296761608), (-373501844616010173476/37095201), (-179720872672814570763/700687130), (64353055350136153035/140137426), (-1191507514351676949757/16587175332), (-8038842702373537879/1382264611), (607927801147902631513/185010801780), (-248463324791491537/1822766520), (-753088221725787125/7048030544), (5237463779536728/440501909), (758080979491529851/212218272630), (-435801716869881/760638970), (-3644956467105211/24466267020), (1043103856394128/42815967285), (2750929195968289/336411171525), (-318129456436253/366994005300), (-2341563606057757/4244365452600), (200615704988/530545681575), (87948790705625/2164626380826), (540245893850/83254860801), (-8014553829/3083513363), (-1165621419/881003818), (-5860717909/165871753320), (4230682835/22116233776), (3903508003/47086175136), (-366544/86555469), (-11827871/445891810), (-450857/26228930), (-1715/1037628), (3948/374699), (2738/151905), (703/31980), (39/1640), (1/41), (1/41)]]
"Bernoulli number"
[(1/1), (1/2), (1/6), (0/1), (-1/30), (0/1), (1/42), (0/1), (-1/30), (0/1), (5/66), (0/1), (-691/2730), (0/1), (7/6), (0/1), (-3617/510), (0/1), (43867/798), (0/1), (-174611/330), (0/1), (854513/138), (0/1), (-236364091/2730), (0/1), (8553103/6), (0/1), (-23749461029/870), (0/1), (8615841276005/14322), (0/1), (-7709321041217/510), (0/1), (2577687858367/6), (0/1), (-26315271553053477373/1919190), (0/1), (2929993913841559/6), (0/1), (-261082718496449122051/13530)]

160918(2)

整数列のLINKS の編集(83)

https://oeis.org/A272207
https://oeis.org/A273163
のLINKS を編集しました。

160918

新たな整数列(17)

https://oeis.org/A276664
https://oeis.org/A276695
が追加されました。

2016年9月17日土曜日

160917(3)

整数列のLINKS の編集(82)

https://oeis.org/A272197
https://oeis.org/A272198
https://oeis.org/A272202
https://oeis.org/A272203
のLINKS を編集しました。

160917(2)

Ruby


p を法とする楕円曲線上の点の数(1)

E: y^2 + a1*x*y + a3*y = x^3 + a2*x^2 + a4*x + a6
のa1 = 0 のときを考える。
このとき、p を法とする楕円曲線上の点の数を求めてみた。

require 'prime'

def A(a3, a2, a4, a6, n)
  ary = []
  Prime.take(n).each{|p|
    a = Array.new(p, 0)
    (0..p - 1).each{|i| a[(i * i + a3 * i) % p] += 1}
    ary << (0..p - 1).inject(0){|s, i| s + a[(i * i * i + a2 * i * i + a4 * i + a6) % p]}
  }
  ary
end

n = 100
p A(1, -1, -10, -20, n) # 最初の項だけA272196と異なる
p A(0,  1,   4,   4, n)
p A(0, -1,  -4,   4, n)
p A(1,  0,   0,  -7, n)
p A(0,  0,   4,   0, n)
p A(0,  0,   0,   1, n)
p A(0,  1,  -4,  -4, n)
p A(0,  0,  -4,   0, n)
p A(0, -1,   4,  -4, n)
p A(0,  0,   0,  -1, n)

出力結果
[4, 4, 4, 9, 10, 9, 19, 19, 24, 29, 24, 34, 49, 49, 39, 59, 54, 49, 74, 74, 69, 89, 89, 74, 104, 99, 119, 89, 99, 104, 119, 149, 144, 129, 159, 149, 164, 159, 179, 179, 194, 174, 174, 189, 199, 199, 199, 204, 209, 214, 209, 269, 249, 274, 259, 249, 259, 299, 279, 299, 279, 269, 299, 299, 314, 304, 324, 359, 319, 319, 374, 379, 384, 399, 384, 384, 404, 399, 399, 439, 399, 399, 449, 444, 399, 454, 414, 469, 449, 474, 494, 459, 464, 499, 479, 529, 494, 524, 539, 549]
[2, 5, 6, 5, 11, 11, 23, 23, 17, 23, 35, 35, 35, 53, 53, 59, 47, 59, 65, 83, 71, 71, 77, 95, 95, 95, 89, 113, 107, 119, 125, 131, 119, 143, 155, 131, 179, 173, 149, 179, 191, 191, 203, 167, 179, 191, 227, 233, 233, 215, 239, 263, 227, 251, 263, 281, 251, 251, 251, 275, 269, 323, 305, 299, 335, 323, 323, 335, 377, 359, 335, 335, 389, 347, 407, 377, 395, 395, 431, 443, 383, 395, 395, 431, 431, 437, 443, 431, 431, 449, 497, 503, 461, 491, 503, 521, 503, 527, 509, 527]
[2, 4, 7, 7, 7, 15, 15, 23, 31, 23, 23, 31, 47, 39, 47, 55, 55, 63, 71, 63, 63, 87, 87, 95, 95, 119, 87, 119, 111, 95, 135, 135, 143, 151, 135, 167, 159, 151, 143, 167, 167, 175, 191, 191, 215, 183, 231, 231, 215, 207, 223, 255, 223, 231, 255, 271, 279, 263, 303, 255, 311, 311, 295, 335, 319, 311, 311, 319, 359, 319, 351, 383, 375, 383, 359, 383, 391, 383, 431, 415, 407, 431, 399, 447, 439, 423, 463, 479, 487, 455, 503, 495, 519, 503, 487, 479, 503, 495, 519, 559]
[2, 3, 5, 8, 11, 8, 17, 26, 23, 29, 35, 26, 41, 35, 47, 53, 59, 62, 62, 71, 80, 62, 83, 89, 116, 101, 116, 107, 107, 113, 107, 131, 137, 116, 149, 170, 143, 188, 167, 173, 179, 188, 191, 170, 197, 188, 224, 251, 227, 251, 233, 239, 224, 251, 257, 263, 269, 242, 251, 281, 251, 293, 323, 311, 278, 317, 332, 332, 347, 386, 353, 359, 332, 386, 350, 383, 389, 431, 401, 440, 419, 440, 431, 431, 467, 443, 449, 467, 461, 440, 467, 479, 512, 491, 467, 503, 509, 521, 566, 512]
[2, 3, 7, 7, 11, 7, 15, 19, 23, 39, 31, 39, 31, 43, 47, 39, 59, 71, 67, 71, 79, 79, 83, 79, 79, 103, 103, 107, 103, 127, 127, 131, 159, 139, 135, 151, 135, 163, 167, 199, 179, 199, 191, 207, 199, 199, 211, 223, 227, 199, 207, 239, 271, 251, 255, 263, 295, 271, 295, 271, 283, 327, 307, 311, 287, 295, 331, 319, 347, 359, 319, 359, 367, 359, 379, 383, 423, 359, 399, 415, 419, 391, 431, 399, 439, 443, 463, 415, 423, 463, 467, 479, 487, 491, 499, 503, 519, 543, 523, 583]
[2, 3, 5, 11, 11, 11, 17, 11, 23, 29, 35, 47, 41, 35, 47, 53, 59, 47, 83, 71, 83, 83, 83, 89, 83, 101, 83, 107, 107, 113, 107, 131, 137, 155, 149, 155, 143, 155, 167, 173, 179, 155, 191, 191, 197, 227, 227, 251, 227, 251, 233, 239, 227, 251, 257, 263, 269, 299, 251, 281, 251, 293, 323, 311, 335, 317, 299, 371, 347, 335, 353, 359, 371, 335, 371, 383, 389, 431, 401, 371, 419, 443, 431, 431, 467, 443, 449, 467, 461, 443, 467, 479, 443, 491, 467, 503, 509, 521, 515, 587]
[2, 2, 7, 7, 15, 15, 15, 15, 15, 23, 39, 31, 47, 47, 47, 55, 63, 63, 63, 79, 63, 71, 79, 95, 95, 119, 119, 95, 111, 95, 119, 127, 143, 127, 135, 135, 159, 175, 191, 167, 191, 175, 191, 191, 215, 215, 191, 215, 239, 207, 223, 223, 223, 271, 255, 255, 279, 279, 303, 255, 255, 311, 319, 287, 319, 311, 351, 319, 335, 319, 351, 335, 359, 383, 399, 383, 391, 383, 431, 415, 431, 431, 463, 447, 439, 463, 463, 479, 487, 471, 431, 463, 455, 479, 511, 527, 503, 495, 527, 559]
[2, 3, 3, 7, 11, 19, 15, 19, 23, 19, 31, 35, 31, 43, 47, 67, 59, 51, 67, 71, 79, 79, 83, 79, 79, 99, 103, 107, 115, 127, 127, 131, 159, 139, 163, 151, 179, 163, 167, 147, 179, 163, 191, 207, 195, 199, 211, 223, 227, 259, 207, 239, 271, 251, 255, 263, 243, 271, 259, 271, 283, 259, 307, 311, 287, 339, 331, 319, 347, 339, 319, 359, 367, 387, 379, 383, 355, 435, 399, 415, 419, 451, 431, 399, 439, 443, 463, 415, 499, 463, 467, 479, 487, 491, 499, 503, 499, 543, 523, 499]
[2, 1, 6, 9, 11, 11, 23, 15, 29, 23, 27, 35, 35, 33, 41, 59, 71, 59, 69, 59, 71, 87, 89, 95, 95, 95, 117, 101, 107, 119, 129, 131, 119, 135, 155, 171, 179, 153, 185, 179, 167, 191, 179, 167, 179, 207, 195, 213, 221, 215, 239, 215, 227, 251, 263, 245, 251, 291, 251, 275, 297, 323, 309, 323, 335, 323, 339, 335, 317, 359, 335, 383, 345, 347, 351, 389, 395, 395, 431, 443, 455, 395, 467, 431, 447, 449, 443, 431, 431, 477, 437, 455, 513, 491, 495, 485, 503, 527, 537, 527]
[2, 3, 5, 3, 11, 11, 17, 27, 23, 29, 27, 47, 41, 51, 47, 53, 59, 47, 51, 71, 83, 75, 83, 89, 83, 101, 123, 107, 107, 113, 147, 131, 137, 123, 149, 147, 143, 171, 167, 173, 179, 155, 191, 191, 197, 171, 195, 195, 227, 251, 233, 239, 227, 251, 257, 263, 269, 243, 251, 281, 315, 293, 291, 311, 335, 317, 363, 371, 347, 335, 353, 359, 363, 335, 387, 383, 389, 431, 401, 371, 419, 443, 431, 431, 411, 443, 449, 467, 461, 483, 467, 479, 531, 491, 531, 503, 509, 521, 531, 587]

160917

整数列のLINKS の編集(81)

https://oeis.org/A272196
https://oeis.org/A006962
のLINKS を編集しました。

2016年9月12日月曜日

160912(3)

整数列のLINKS の編集(80)

https://oeis.org/A276401
のLINKS を編集しました。

160912(2)

Ruby


A013918

一瞬で次まで求まる。

require 'prime'

s = 0
ary = []
Prime.each(10 ** 4){|i|
  s += i
  ary << s if s.prime?
}
p ary

出力結果
[2, 5, 17, 41, 197, 281, 7699, 8893, 22039, 24133, 25237, 28697, 32353, 37561, 38921, 43201, 44683, 55837, 61027, 66463, 70241, 86453, 102001, 109147, 116533, 119069, 121631, 129419, 132059, 263171, 287137, 325019, 329401, 333821, 338279, 342761, 360979, 379667, 393961, 398771, 581921, 642869, 681257, 687767, 700897, 754573, 768373, 782263, 868151, 935507, 958577, 1005551, 1086557, 1313041, 1359329, 1583293, 1603597, 1686239, 1749833, 1891889, 2051167, 2086159, 2133121, 2156813, 2180741, 2327399, 2364833, 2402537, 2504323, 2556187, 2582401, 2608699, 3120833, 3238237, 3557303, 3619807, 3875933, 3892271, 3925069, 3974497, 4260917, 4504663, 4682791, 4846279, 4901431, 4956893, 4975457, 5238847, 5353841, 5373133, 5411839, 5666783]

160912

Ruby


A061890

第5項まではすぐに求まる。

require 'prime'

s = 0
ary = []
Prime.each(10 ** 8){|i|
  s += i
  ary << s if Math.sqrt(s).to_i ** 2 == s
}
p ary

出力結果
[100, 25633969, 212372329, 292341604, 3672424151449]

2016年9月6日火曜日

160906

整数列のLINKS の編集(79)

https://oeis.org/A003687
https://oeis.org/A081478
のLINKS を編集しました。

2016年9月4日日曜日

160904(6)

整数列のLINKS の編集(78)

https://oeis.org/A080790
のLINKS を編集しました。

160904(5)

整数列のLINKS の編集(77)

https://oeis.org/A051786
https://oeis.org/A072713
のLINKS を編集しました。

160904(4)

新たな整数列(16)

https://oeis.org/A276267
が追加されました。

160904(3)

Ruby


A generalization of Dana Scott's sequence(2)

前とは別の一般化をしてみた。

def A(m, n)
  a = Array.new(4, 1)
  ary = [1]
  while ary.size < n + 1
    i = a[1] * a[3] + a[2] ** m
    break if i % a[0] > 0
    a = *a[1..-1], i / a[0]
    ary << a[0]
  end
  ary
end

n = 15
(0..7).each{|i| p [i, A(i, n)]}

出力結果
[0, [1, 1, 1, 1, 2, 3, 4, 9, 14, 19, 43, 67, 91, 206, 321, 436]]
[1, [1, 1, 1, 1, 2, 3, 5, 13, 22, 41, 111, 191, 361, 982, 1693, 3205]]
[2, [1, 1, 1, 1, 2, 3, 7, 23, 59, 314, 1529, 8209, 83313, 620297, 7869898, 126742987]]
[3, [1, 1, 1, 1, 2, 3, 11, 49, 739, 41926, 36876163, 1504578225617, 67856786028033600651, 81238311359334144709516343054051, 8472940010945536421401513734595877223414710434640386, 356342719455501014965525933507810961154502928933770839954283399616194368420340104961]]
[4, [1, 1, 1, 1, 2, 3, 19, 119, 65339, 67258454, 959259994615659593, 171965197021698738644442682357, 12959040525296547835480490169418622922155526267774117749963303914461, 13002155809155454693445261924537731852173830380093507167011278151655802432083856288517653948277707791285132881, 29400523205612366107661002554337110643721041759414946608515448389060941813247878464878473439234370680824532331236610288800098568671767212877537207573869465586460220343683206513464070257487789507650267414634044677459465649664976609794888630168043920806, 166196128396556922555899890672338321380534576774435051327037692878096005394631694768782308023612125085309564596624729352466151824283077070922623219055586733813876777561856375854938714493491351042021073886400413284004094914487033915599886441880275759461887287306986165428867952288409212751752662721141988978946890440543741234976543897251592566287866770705851186535438538629125586598681867436841180692335311091]]
[5, [1, 1, 1, 1, 2, 3, 35, 313, 26261407, 1001689887346, 356879751557595054813966522072161803, 3221974575788016845202611315068840860244866942009716269469, 220440582040369036916185449332621560426118324976027979809456433292164687349906793512780771298704684702949036043042770626334411128890815722075405977265205105522835656640731, 346638866838896819127952989549289053182170576355992754417491914222528719453954543043906674857716621289650809767744831870162782770943990042780704899654891615726352339671611131233855088860740386421232017540117240700273151063313011107033014133100380078872236415161941674931973027, 1458598750944912333572324981891122710409675841797185602322002117350505180081635442004734686760556445360903938523016747868890194046423574280014580570619831094087570397442350111020806955090111860942276362772079030462200410075991991083928212080196958733599054765263194210454135335553430534601131488447127689242869794986150738631188798681140930952058677181010805436899871361830065543297330106983858773784152339990078261951767970788252577473761088900303739403924458163186683639576369853371157460535393058232836502423824541767653154375463239922342927676634020105474266948951839333979654557191040093176105569545533291125317508935074984660300204869159280882502727050511275317832945922302745477864467457417348543090192721162032190322917847907484573160327607552121697270755033999126848990441748771781388020726125919815715145638, 1553331582571901059742820400301495776221273820311671687346779612632413318195141079710933666968120448493790349516276897199015500805785647549822893598487359442126538822695634936135435941331366359401056873726430073230771122792348566160532939733304989133052011184364663388867003377536884321271491469019485285660121345406948212337554192882204242158747788310961364643273253397697015067401405760010615632096388327827180207395885479799144049995863958414632519049205935353857247871736978853865739754067931807880142774353240724358006120185836319021302930283377397949718404779458080562509731494285622270638615820916027795542036321571272965594489579308119921706287059679794121005935702754283535047015852434753547270883398696333115857853455661092259816991251347026787018591101275339001135568098092838638072905790506808359142364963137775649199089115489164867997830065634595652333026367702829521831743289564490780220384689120247994951382938914779518374093499131476901633196622603851124906262538653370600726647494721953095864754285093013651915858575453756389354537771663008691463903576311160460429569842011880539734259758801929200502678996687748751342628273555629571942797481104306407460795944614345787596422966502746846630279657124730192260592500755722922401700418966727343738664352056818823125859412322430052735505706661757412476994265]]
[6, [1, 1, 1, 1, 2, 3, 67, 863, 45229192379, 137704047426799334, 127772736198746335804079836082474464221836557932990728080773089, 7900770309508995071687598423487153335856893480509233473951472671773030001497008306243500805487545669, 96207811984419848434465196538178559312699757432314736653007204725525498615455574857264471726240989381618425908554249557601757931217763515995653740643830282409568468696743959310518070748076470007499259389481270357827684362009647024837268088141075128154215955270158780788907519736609271316125611450439275929641170452771185230278151999709271419114421394884288269133, 1766322137126313791987381375779018445672750436646051499282928077099151816976947280457642971831703161139713635040899801715862705661242292182854388255978843239721376740036101021222414767399547455770092996590366277953062164954333081248238470822678275580113921142297837376859591092979226525423192452430559849410432079014529001674398140457593208289611537683954472217243652004867327034336377082362666554684801023298328103798708267384171594147632539448408506856234248169224925584564232634446135388383239298681424488319116867200859632172807500068127461262329598005930471421712734066780193977, 6206172341921198063323208265228778255979183246784946921248451339357873160839287272174865137726336170019263197093633653684297645342917652632606569382355573408558190469152651222774141904852940975136722565508196884275025286930394425859763057442494182532826317490137124687265596256902491227274174945710779634888977920799498982010872521184229973112588032364880117195299710745529775198465207434801161231202124395125830588550419168418470814312484718071881468136333774852177970414468220022430616921220760719187413841565395874473592348113338065695654933767077497975351683429235842850020493102344355931165785899003873299782938199727959699734786509666899794700317514291580604128092258932568820275539401976594427998154917792824718261151309953424934655538219251036541196834636057525923041447807434353670862498275414190476416032161410241197880598591696740987156803122343476724413575339717858145609274813048791647452407631788420987013052052571544475370125880622931273220322248318571416546210446454557483136788443005783090941222114901254893631558747995575399783951601526866373889981398536454458041107069947624935630619870024835861164003325034624591903952846561716475719893627791542692114447002819175210642468562116941201565179331944251012600881448101930896733687552881973198907395892383779808057246297533998116520197803088117604172024272610251109301335328643286764309531063382447279671073080089942943533013144395301918567395845643844532911643603161701037257845943910092165974204098984760321635156687146928713911333014388424456072215056820391162199427054407139020893312155008204058750076681435958066319978924119977929830183512437703409040029974636056718854111227690276239245023891556709267419252277457618032083086171073404769811838463715796339072878508856349969303401858006013973229623119221941220132295012935894158548806897187613154901043010210705378847007327161148626490905772399297049901218000919719421285338841525185324547606348934982662183462519896835648117191902149125629872491620215327360666247337269194015385585241780623918721545491521263513897888625201369410925654299143018360151335826551656989146226778413996249326038, 3843704882633914034173118730555813552615333304305746358423869148404326003163246536427882936814726948586578087421833719666541238109399477861427596779577717459845942050312927894015366762294148447019756919174339950900836719176284910791448339461421805674381497669591135805402855995168100785685876530787189641292547615586150225586299510979464530867495329703811350534400069720444104829816136239669270781129347472173585154705035080661535003680174654038813239147255628425488166804263167069215267271361069782440463936961232963637713942831953341697161654872928126625919413742253237822420396641174212092912897109860205192598403701051701078113728892988393445406374398564214044285519510633900160599180696311556244134384787353655309003468286418565074766315577043872926621981230313825980235852426966869640266807165876739036599176597466465416101548214021264172961187476967768114287022342621906878440221309955244341034897193221645573218813400796725704435104620778199490862921887370798878842238849475925630814123582989066416706946316882438881878154432139969359885722802483486848806253284974309425941317190960373418772257464480480248672223248093735775741755608894171723866168436662705803594505301344300699855256057917248588894999855148794553217145573327146281758022822368181595725624833640407835309553294805414656380951761565990578265504503687511446677906895036969852768898472183315701633745903999672580079403329506843969928816003096766726331073725296791197284716898474998962437843433586576055543726202088735852660641622694684022824671464571369971985021312163262822157896607855377765814143261087957430227249523875455529509503022369299442914191203544602720220520716791110805448329998744957697388829099268210176218090394981386301040584912653186750150419657728619597188784576622525967223694716538567858877961967964711871177973661092839585170166585423299917029965635298136987557298606028212466798161341997154276123016903978035610497984553272823813390710475062432851673958126062648718116073933705852324801909787158555326498516526689103882948234780093729493724074714079424095216168998892703040439276659823695018165971417826544232865790264846756927475034993215971931559546602437932896644353418655979110532333912089544802257780933493804138168410726559963790054565679548877640009450471942381735548564964619947624454799628474625374692128258531909445965793788474718566517875409869279487118477378044637629479587758090754047271058518886462509943832360190616678712624440058391565597925578496558290705248283593710112424280743675793879748942456338416811241166225862418958088615507758456482521498124161085535348271280315663881866776762131900134711250371453304345029189285504657545048460286506537785618813066331963857090878133181634049679294323868728147906037650766272979356725208280713030879916854062533416031549110466157509188711163853191416910101833885155966395596919200850126503831567488438141257005851042871152634299997172027083211133374153920935623649930731178126230363313161274626731012404279312198073176353637321537822073626123800541535481067711717525840342044578165846474428741640674832959393395068060708786570899265087602146468768466555459012302489286659201451946469864519954919216469671385139634436477081043220052417723450965852800941713723400243803918660550168403508383451581202876492596947084052043933497577487174135260353679212173029137185607773184220419298504159977106065610732989713545723340505027865797538481952547]]
[7, [1, 1, 1, 1, 2, 3, 131, 2449, 331031310954079, 176116593574686514024666, 3325157506636323856238853021029893256716248880184368268505553793550581378156071428315346547177546803, 2145918147329102955986291632358938672444415223632327877055440763383420365179491213470464163378237121716528996508003832934569812265593136240991815087965655868957, 13577395819728575966896253476571705836324557599079932980214753835745900944870156790027922404693270038393417497384964535333087716438114364301281177153966936405076777949472833094523181913640387599644712716890672694051347967608823212535506196799335318499532687469661417107905746849151899503521563390736688712999404853016555184397653270032772719067524898005337032797202632577619009388016277284601004424776997827019860256694506981588371408928316196793383619406648404169199386192822234067438272509927500444057480980669911976235750465755595473454809507901011026942509517485605191609037374652609783976884579088503964548140602547901558162210215430914126043917861955546044577458779838441280531, 1189848837053118390007666624794575592489525073717246173394775247327446389290626408714064416991639693051227006060525977126099066241156879324234889602908974073048905937065098284984145002362989532906084763905282315244035279248498595205441874685005886375042650659749895506128625470221075286971804291687951022421599287177060697707222555103285219733792426666332261566034329587036518717709829239458442188219625812114960753755090215556208902263222504943544381239095328829706368628489603861580578520012503131160602475750212601246114887729222467659863171336801404627142912121571828166446949970538295773388356027517525158187626300116288898379697359318162527977340205683584624225211059414099179767008713577971654423761683515691377438569762991391346376061445608432887146640857395150082790662142128968458843334395883194608996482517666248879482488058812150032190153539658756483968847880639618667543603941846128588613339134601500061818932302108759895215858310474833186594364708404317331685122872162935977862758546844273567191128821924218360249192270323945793178261035745302425689185174944725647993788966654571, 25580152654361159109342423403525847770503170976626714942558460859459938056177744803380121162157399222893442974419789941899906127905613233590626829588017313020373444225166036782624171971938367744634542675157849185183997753252505447786339830220468019696716667086018724163349154982883947176425904202784837775713970908310445094482895507062293513879273674655930666149584609619374731011460434354666310887598417781848872319692885826913276204076615668892731265509269939640729145258561842873958221947476043912950658597989132663169384618303552458751384242676910595130235141226447490241901302538409825197275624358843931227331210063687207739137341370684330612422940488163377190198206438144430142107481918255542011615260629744248966324546401911971588641424234837842815189015674445731170538544770108635197223596349099802982272500566065283112008323936168719226983926244652536717704061398541309811695941814590139556247027584145682495185610081288854982756768009869787345190830188006573989044762482343863774644134005861426553323192803697634053001748378420049633744438714236162896964350983882364236271546729666684742096852728839308722952359841334990415830212376934409102074799078277071527275127992732244746557093225840982877495367296270036951367597646906870095493361303497153471330704883386243523059423986818931330956350831726248854103643544107934371764726686575200636590279056509084845266340587600656353575276040981280006520392353283481986555032051019040703407009399364649134700297596977640349159663647307895758139010077374537236833323713100268356004526405474577239681442381796571924715106342764574077460165412710690701419445167582233058493984784742284636047929869118178818115139900341330749372570039966489991903950803374832303387634468559870750759694963834120921410728237788471621314236659350333905042193567614289734680086893918002707630162158368509964493906520074769385374815585679610970484955282532116337400910480626759567754117698957657160335453355012091637540736484596119954545349396027911725009258788259804402038894566066291904111679627223921150722061211491206801375027425358436436521874881051833276662916081057587996899885959100850432848389989138715204868629929706893617030530607532727273788938778768966388080686197541428350969803281855689148381358105715100325128086503625671080662224996418280972407804313784448157913294657963959965998627064620896614628521109871907342459541093242816229673289821180617802441932332245368717652352743195375254227368322276489596597920639966908973652252054446862535031950679583711966288817602996280307812173825920429381101622290131982536860471645001644511025209644209233464458660086013059479113839332940777850204874701500980271893933232440285971612056647492491390058137084095600238014036807390229456352090945143641125673970114146387667046477919889354221277149212304278443624549826383393520255603388272014531659010640317412725658282293518904496706013067224160634248633445944024557559332286508206150739162627912862485777456942477942545666515553811027542305571103886619903017563567007485006817455640350177604857014584573964779617512312008462862021197536168505453886429087921620787367913646365362375487139019346019565767280455036124065192941993743782376856608830460084190834673260140785793389569322094874624994557054421578683770587129983770483348348039740686430252243916740803950041467823578181762573209139399246100346677871814297106036442016618548104900511631777063348530671974877310783281321002767686178084134497799593233702689975275982209425772139670233447020026330785212798944459801002179610628985943786837892965329872995038608377776909604292712804133759675433044576764347377588281084727844682896760487547089501072210832913115558252709613397578156615420595445278845569509027384450568442957698861888411267325133866943385061226718235108130144380074431943030317646341969431778995322045685292809417817375390858870082650333665002204458649573904166951860115292238211966664732861237187483261762054825699602677243511482672874635621746953088180931819479306039336492248517234114064354103066492510916173640534942474132265267487345053443928054457882938831559197766512309634583246402674002841995854004677973923506514640722863741476354071987916730993004502147989702712888113962775680945020665186365797092584522702253818783688471708934179533091095808587085500688852557586686071610886985199724104824075762281626881822414210225214650328765723275238583345700186888258320715236784372191572169191136794984712046243278526918888075496442667115877799764162688747280808712240859590994243257933507620193134017870509706922075948746010764105221003776349304622999158274202641760338675867771617737021942773503349041766733305631977382490645855040866065156942337378248053572712994260345086, 1573364622743265117476158333692087158261029307723665054283618799456710733913135838773511379943462524020132742732275600646631056286729540507531717567892197387701055776582215095593353796548383251712135185245381070488333052075161605440798842733548637002382424929012300218885042720274353703522389440763797001837579606538577658897946658556488699932510324885863648990518757632205453710844216255947732667722054675154279491918012744661898453394741903781649474475396132027950859183850048594182366445988565667262419259752835891935788756426394557213043413772976058381439664583373560749560708373325170191549520165878218430975496599115524519569606438551357343199757786594525321092961858697963694083025481267784530554703143421272157782465677890630462284078158030100126163266957222796683575402210517661447066290944451137991505221606252248545153059448412517566775323201464498201337372885824516864993661990932633545491717917849892192433524382229033169501161843101468920154387658604236233755089663451968467798229319469053140919104409937349530396247940152257582478925876816357070517395588472325803819945351601545605045894737910002565905055382648384775881486269919660765194376185521936089925122123809000866321609546211784347265445339853863358515425886366706601052120613546444928514713047109439757736961576100791369715890007784624203457685030775864747282001938172334157145602475534500671335403598051056800380216597802443426341874008089424783785439871765362851305162321082938230938650416900719579762521506713615039409037129745098662937876537782264535674458426165886760367387437685737017249092155006000668114233754438069247329617011799423601190150934708995078962348425919732596587821036552350881059626914884193790082576543501923933006026041009931338722885717569712220268705435797833737771245525064234284927662385753578201058549702516830346158856594772121776888202755084992108481436251243026003718998450464256337695683779764617213516939875425349877469301165163635815391804443151246343379256490481140259866139308139421130653410251082937497886785513345431862207616749694073652659952075808943920563654910923766624271452329077188992525552843556938964124184039859691109151260854774928716745634351228047992347802721097304403199737219831634034567967684456456130149470001232222114781228301102507100367414792844231975175995754850139528508309892067714025084682004182226524901045638041683523229967179902966847438446329761672812598747711738057676097967471213011894199508090912196412702572286695942790641477529071675699101705867942680597967625174668809847895901209640118056602061524105537959564466943226438974066337218316163625971770281596555825481550572092948627698871798721092644216086353315036259897516701720848799183855884290796149246890183333409696052647197630300110017143468067293737058229675712992526669282896142839859943110958669489534507058561223497772761296995833780003381691648441023551918809796256318770317682860565559167103895929412266952758846109493662239785582314206270986604788699368849945896932318332445103206038348797735211092515745310173166573197945644651166864943774585445094262008125216790679383390808321228709088649633464159895220860775588432695379757130194547858542564594633000950765040798587614553589472226108700506662355472778851133508159898022782354985897197841181318088017717246309227442971492474475906968421411643009417271775002577725216829559108705958702895287480348785550805204776820909844825489314595070546090299727338575550377588401817182810609416504055061350647671016850689047876654688277677144517441965168011245305209280228546919387684577766575193107926132605476557037999340650837003292444218397751931672374639663601655406832083146574717189218362627801341524302168571607812023024418118162950426542783613315416002743285398159715978995749392292311982091313987082151141129323575880925640379398390173061722676082728425595746051069605879650967744659149443256638074232843856683929360780818607997030920375699056778095387390331730002439164190256891393762096764590694347253515513901507743652193991671853552888832724571539361124770164501032403024181196278643933868975095499001459149732408149497928164957694096245656102622353881154288798097527577738439413077483388695355702343911817557004002364040464961564231477677233688042992600963655163466957985629153277838940095789218015356658018087965829528723732881673975310944107008059191353786365897755003820708183966367509383270326265769760185462539284906944035344339932739833199999719631517722275380021176267769001657264416125754105254022879168543803743332489441367701420250280581932047471638406067078432103775050468747331724917271232082028684513590347794830445242448271359412309810028328009215823298558650658306131768989141880827770705387071401297973295115169561142685008937538444178175323322371060939169367006513994552900307665711088001657434336501330862754729415621146530832622666168455304115713522121622190883201265250984499233420455216749607863562938373558343375400216267560953267862139460612003969306698368109352589490357313148121711800980553838212904471745833607445038242790777713989649583355776638479687779260917934284878517878575586649125339994610786792384763924396460918089007446543649672920368181681509404782965351915410355170215961391529225246305933881320646475624923595681542503267316681086174707127085829397209986559378094051059849296833593166214005429537430636929602753878708309584422034737400225108622643480918637629539527170504419726492007943259533806680286078532000448783116736629720971068253793947220940706382435130207862715107200031214293454514024774558266748835798535494357182863208827408794963208560457196025561378441346554639795699016869771325217914356827909495035584261039415107863868863546411479549072024670495023067067504683747345710717213915608076717115203610338248478447631186201306117072042602015528294115729052292989065636907534864083721452357669979342482454897400410016040772506564016694329587980261631912817994292156321341612583882133010022245159765280884874876478127463488822612510411363873698256783745486958520764235791528891877903782782584789744640438929222171277631593943972146731139004519547689321282098286477157212721380075131447213012053925064751610378991226083667723377180667485515380086870572914127846503716667711923621750963688684230181642304435367027542245430951542131701059607654398188610568251771372971636387517053628315048775787171646371378362635573016429103641103896380951108755130868731000117396882990875718616865038915829965621958841546245260522075696059787340117281013136359083616493407131859565215340713314444254650640360046279431732184294320958712678087928378485487214423938535906835509212196826649097189313394420800615227996262443651956475212432178597889891427714545535012357852157476034942963339965478259281069479651246285514004120305674675190933820321063562824021225656853783433952046726095111895296446903995096846846441097929690543947538102266708025935098874149980490926309362473696520098937302083268755098771549517801315249705325432177462100347749035299195570922690070601705505689535618441035735247463000777884279766858287219282399295132092611445378169564700192335174580168782846488290644745951217209881030112885775195350875951359476363867474088452354542460037263283270933906642327150920184778449448925826325513131471208440877562204809470229977877319227304487511066948365878969833746485530209361933876257781299840464407245450416505931751601586394333956130470550083438873071585825677645525668878054770244652822357195404196515018717407704781732967337653397944951762669495923073871780684288980517808801]]

160904(2)

Ruby


A276123 の一般化

m ≧ 6 のときはうまくいかないようだ。

def A(m, n)
  a = Array.new(m, 1)
  ary = [1]
  while ary.size < n + 1
    i = a[1..-1].inject(1){|s, i| s * (i + 1)}
    break if i % a[0] > 0
    a = *a[1..-1], i / a[0]
    ary << a[0]
  end
  ary
end

n = 18
(1..17).each{|i| p [i, A(i, n)]}

出力結果
[1, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]]
[2, [1, 1, 2, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3]]
[3, [1, 1, 1, 4, 10, 55, 154, 868, 2449, 13825, 39025, 220324, 621946, 3511351, 9912106, 55961284, 157971745, 891869185, 2517635809]]
[4, [1, 1, 1, 1, 8, 36, 666, 222111, 685187756, 2819713283228248, 644335913093223286486628176, 5604757351123068775966272886689217889936356651, 14861563788248216173988661093334637018340529129342104300621091389266132702213641, 19033940748700075095139742840549610201939372859682653242963041791054289475934606662871084153376369462420916627104078373445720327061217941, 2460581192134651587756067412475357668253126081159713921725620409979439048150817623624831503181769998065631297487087195728430049110912893849387918348135642795967650846588294536864601785163769717782944361366497048669031151647913539889353, 124186420049040628853505757292121205769069533839325877016963114875096099963083842976205369826871166271767629589168858320975337885692588961377762680675626435153855794020553681493441077273301832601227449762842711386133535158859023606693798276686316989735863895906038157291088218223527575201383094006030471158042038880132815127604664772548372647553419176096537096649393876524614250068906593531373103790650056, 391359617594846908618957214312765316873383731761601038669734630670086686702532545819238728039569863757575055478036149837489679865221818257751919897273920466857736778463310825178649845627872241295250305392905363199223139886515193985736341552953970885685796791612110489453257162135093502724716657112887157926791349422486891031356674480270569737766097190914588440211171757718858195509226935463043995788024653460493920877274243154187743226833331172081211224747573123540659482270183680103887365671046496250118054380045943816442076254416056171800815795086263660135487866648873592651058782100903330635164896286191457498208117886160175389569648296868328701819933850608919266521093728507724335695893964636, 6282884930405575225029495974101387723317906361645833379779062912167663891342870129436272209617146245766175072646992770311039789036848648321884437370028272597052459866219114708138473429100978112897829960121546112389634438754974253353115366296415867970456746153055912566470139605090381175305724634617154716658954104212381856173956303021127290973436730840952562212580849926173091882574269519739314883566936806324171593467053685173077904164056266013606768213023571351870381150702200050001087171329355862088908856387920716349185571366151747586373140793052809068547351634947477862353920039090789987923829382696190983773125850604764235754335672735771048560319716436953950748199645051782581604929701107919053469647475822017189484471459540047179791626388954026981179972291358893574623908086527942132143098344534303407421726635654141052587967365734421990369121280570634427131035826312836122193252139258054035017340203329131245897585618973192658114744347857099075778778082046339942665631309695174644440190590114244986852635759843804815386411136620923967546355269488691546924285765319849221239696617343205123583347581506021738952234271548645089025156778277628180491416336622227960139818953186394963234130847146, 124099926550392496669962758936124926182825500616776938293264925678978666596004993308549227913716977044246283573892905240226459508777477058956312676229864025285645463025198649702634820498160891939173248050067289037079033899055176623086635690048485634670688061589589727663837620714563663360439886773563251936646686861454381563518415269593756971651170330969623892274607320336847233146491472131124500836245369827329998574360765761190001271787815952442428869194364314312140307485008501660334521863099540578372507851119889174400632409683489283106939479277566610381634718293216403120383675035930405274900156664285797429107022423048669776715104434363618826002112533673047649893468428340261176143076041817587096526422726769925844566200358304700127101851924266395479862274537357522260707949982010263741979894488856422708561989552205104067315455787036127449538658346276471588864914929904499473198444898937858440346657544674505046742298640697569461317684569323791793326081347530709031404263377727169100791093210132293188896614541572048355823520861656265959943668861323461196193876513879320352323438657149859753751333496258075373295235693338418807157965114348470780299053488873237568027069632743995335249473351529970215052790414552673692685653672898694244461301038701143153469602855564446801694180523756950056479839079556060318328168727599132892673351470607417622017440542592061254671892214997844564731914159263743299996751551271656137834547202816084592337462859196490539263407901669888408494832689013684320466536068654666448756310136491344581516928738644826451366176753657133203921199678463350755266896992399794215731028018650801191585807461019225991964362431301143399783250384830550945874782874692161893775388914577070330284974180605763699654393598025230182211799015627716956421156580227757897119452732127803900530263018236761447117684848357699958177022854009772520645188764909215969903107545082718165219603729554011390670351922936708910542066742643175562918343814344926440321466191489511929784792671760973859647862118450134680900792024363381138355368081444102780872070300191]]
[5, [1, 1, 1, 1, 1, 16, 136, 9316, 43398586, 941718655098991, 3260419210887695990745089401, 9128667609038682887521294605084509642216623603458676, 130571418098949527667701001911034307254031798383845757340189514208252911098355766299877310718094376, 84328534350602708894337659573119881832217264126095729892331201940528000547706833384042063747288964196431585099881272363930533221692235206131528298892918867367494839008972429054061798576, 348002478278709886416920436874022630004318491215316808288392367031301764281600346014585546658765372408610177131099240202718829088621684059438847679377602511778708544733573850360986496105230665009604640354269767651529672287465485556359653177653248700754527321375640444285845799107280397613272926906439761489017622716210558576628514480047939323927726, 10728498885725775693280651023434025772630798108286732738083385097036184803376843708861925605463151182885413082399744192260524526228118873169252772487445015529024129578950501101420207913633386727397413085851595000019609870542749721391661716907193218821252899938324212931133252700870216088018773451181622673268054341458041168396542727132728488873239645528790821957872372641013825040571780208459934696322121417818515197918565426337666967631946777907913694398789474103307134652582493824544410404663571501835756194079548480429703752334778864420147158746867337871921341111991554325051249379033995321767528146902887606067268412987789627597703439211198081719131291, 4503359071036351373606933221406223134536077425180614120484590828176301868520757709623718005302914702639144547684537567778012018011637416845437236001915284084405504247222414892197066110428575751431536008399682818045448668491986173578121961969019912847804185019302636996552430387467891476963905424127304707893063137923574953890530994840384325808746444674544089543484536037224659081154196999871073210517958282382738727974348345696467844754363733617053740826782961820824932379679777882642753035506897484475175724318873532248428782966391035848965582891361153072731122331602878028432780953863803205086879967681813022685125986266049796284360273689838166889259654027047692157770617423816894917663812899063975120418434312747528767562655050933835281131521850089636880741903676175291196788533462450777388973173780426145606007626057384415602065868685328934596757577234874214501902281377062302429505863666053137186120410414445439454575168058277444307293088175080772244925704382493129491402833479974925258877727072734376299706967276233568147688898169998566441974103909453253085441557022408444691302498789222818601798968430891420346185826880334748192671108170604935151666886339616106992812751500574168863313307487998003936433564799050377367728787061, 10858861775812676343398477222796159622254559304122665001018644442392240303321907844614989268463482191934025346517060598019925118491581219798340142704424823215290425525907252022596749754971801664886226813845205292351318250602001218564982610658332552172899182283090734711259627462397099214980932705093322699264016254401879723761131420974511137047264752806945372063459483342488812024982546460087115173953127291356192819603772317074709202730048867949794588277870730260538734067632889558243037173995133493925045956299005063282719519971896709087551166736668853957970483623118924245942588326426352459180809990570491564465948332676075402771852755692043650068261144867368025561467682691053048630066525386832558798930585738331669594174679087631667401082995647934023948397689915285738207436115627849479493693016190298625225675081176185562216022239279044728532363753358442249399914786140264475206168227457435086192681116603008157833776261841604558640934554630587027908364754424665557027776881998611231279755656251526919621410910996708790499556151766295988373503930521088323424960012889013097905654994785631373213435637356353049218158338008202789500812016233865412619577088035628729500126075136939045165136071355507462476329340071762178403150143618175815417432210319371174291886899482607686571389525820996746992593205269210655109107040177495173523528905631716327007714124230984950366728653983012121156064937212948571820979763774698318256414610883028424648383717567059318387081476331414512172502045424778705919688718629342944605872602838648112587153441037371316799069713325650510940345165548372294440593382679976009367045644424398899109503776062524159883328031777644634118138187855272913680057039416773549861246061453130296394379207702641651065994620860912188190536976479740297308915204720559521264091267636796565806428273627606127562344740840305956612081193574232221529787490836504928662712059094663962523623785651472509487371337983480945650479447330747913878340871190358044809390140913604349741622563915711770241144779236915365591605913210190119224625938101556890202680528826294661366610938596870993281050548110201684011549079668060197553539131058346165126614521656551940312213423174403431540541352879024142048985123479385120707424841348844150261778344180310128043529689419373292489328982364169843044719597016104209205464015529477001191, 2165048483634836976295968965773047647200423840592225782629946518928957385515490890344797458091768983660085084010859984556390948465079439748089206346283973422325839641019046453296791558085109055437822593079905083336929115531716787780508679377138831444951498792681658307457046735511849496573642963273540835344569159382528378032981323440230343334459125248756721225571895353592171191571929385728508599921337056272074805587679986726849870338290770612698127242727046897843847797890143664700886452464987078284605598830452045626710890770964621415262798092208865685828022350447462138441325767111612194424927174451437303931065095872845888440181464029583093650477245974548214722217697830900337645677725099006949384914676113291530104098438785304797353871134471011949526512904987737455822219803129611387222776357466763933292712700360180929917734850044654286245867338942715400816459774155097058929281892844658190597529995596075438257778745370156298187866452116755802615899042933195756636747385025752956592677162165850347070405886958771239847827853015770677780970291842636305733951357541969704107708379719109071604197508400534344612043958392667408869750924988940123338621568601353721336256680535241450849041410136790116286349892536333031067495239392529004840136031814707667448199397621038221277503298445021219230872496655837062433081733552200330890016511291673561111920930899739746370463638390599427763973593308996979872886264122796528717800500041555187018056179990983021829989791894270208219217913384816579144212897227734043816013248208381205017574522174858395042609837668840895823745649844286707512999424514937675146573144414360284303889667896204888230162483919785022640101594827725063417587709166284018317502424673975960263958649275345722096875288229197573409131956666078054819086006355278152694538770458867685264725552050567264806477767202064264891690860071112144207954035976440369688491763617287578056202261666884530498939420003530153796828891816957845325606142034028699690327213700063756321443864674943707710157892001223696462730105386177723030063267551225147785358281156824351258292757084960755544054462380182557328587003814242858209232590369771037975905693019613405913912885696219854665549257959007393009535401041865358147200518403479557796629558833935399608533600295548655120719057716083211797058572244781128495109977773093498916716635131821182789828327594313472070211609990108869656263785662407700205841629769653429383039396278861969842193788204058319064259975672098413256981878313546850808438649446277038571333160125802913153673033899182434570556261234013594443759286149018857482612746536720450728809466199931166127998645835280226927161958046819849445090279982657966392880568386861288644985663970155952479759388963694650250624074456593607591764119489361356903307256875929808731783158810427964193276660118593934677797213972354999338529404101578386729853349276146801874773805135897747619917139217260332245274679308333423862067088544687199775686485871364707538547547115428003072708224657160338978754667705396124667221107267861230601681997278090869123278230671314903440796736949333264616617029974563513178142220354958388215615591963552257349693065231234604800007544994032932418829190899503286142225303305365869954651460536327332855177547781683375485080798432292961248702789175618052086639773860289695037971326563029235106805000174732545454841848163148836329490167664804120864742820504921330823035369539919471095592728462391422460282442374402282673008285552519269927090513781458390064241631472649746496702699068767477540834167083163815818417804719492723851195278865351525931626967790051668142900318588849669289451592665090747066644171055609186983145229141773621833253271328680539745740300388509178219120610971136916213710965082176522984514742645267719446558517677791644105447384613377709940627122642826538049210690773030948237940471235837965861090728333352102798654662495234490345356017138598302925755493823417565277243626958523085635917509906309222397496835965300828769417139241088683914824680125550827279411239395282598450184626929388598004832486284600020868880132933698211444517702736551846360978479793972489465229502542507923061353721716504674223587277252110358155194002009216353719774990500924887133020905602317723261593662484457811055587132398968162587044321067004225778744499857270530720905367841730586324855978528790152516697898131367064172627225130035671885887227500332158036]]
[6, [1, 1, 1, 1, 1, 1, 32]]
[7, [1, 1, 1, 1, 1, 1, 1, 64]]
[8, [1, 1, 1, 1, 1, 1, 1, 1, 128]]
[9, [1, 1, 1, 1, 1, 1, 1, 1, 1, 256]]
[10, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 512]]
[11, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1024]]
[12, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2048]]
[13, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4096]]
[14, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8192]]
[15, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 16384]]
[16, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 32768]]
[17, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 65536]]