160112(2)

Ruby

n と互いに素なn 未満の自然数の和

確か中学1, 2 年の頃、以下のように考えた。
（その後大学2, 3 年の頃、反転公式を使う別証明を習ったが、
こっちの方が素直なやり方だと思う。）
例えば、n = 12 のとき、
n と互いに素なn 未満の自然数は、
1, 5, 7, 11 の4 (= φ(12)) 個あるが、
(1 + 11) / 2 = 6,
(5 + 7) / 2 = 6
となるので、これらの平均は12 / 2 である。
よって、これらの和は(12 / 2) × φ(12)。
一般のn では次のようになる。
n と互いに素なn 未満の自然数がφ(n) 個あり、
(a, n) = 1 ならば、(n - a, n) = 1 なので、
これらの平均はn / 2 である。
よって、これらの和は(n / 2) × φ(n)。

(n / 2) × φ(n) を使う方法と
n と互いに素なn 未満の自然数を足し合わせる方法の
両方をコードにしてみた。

オンライン整数列大辞典の
A000010（http://oeis.org/A000010/list）、
A023896（http://oeis.org/A023896/list）
と比較し、答え合わせしてみる。

def f(n)
  return 1 if n == 1
  (1..n - 1).select{|i| i.gcd(n) == 1}.size
end

def g1(n)
  return 1 if n == 1
  (1..n - 1).select{|i| i.gcd(n) == 1}.size * n / 2
end

def g2(n)
  return 1 if n == 1
  (1..n - 1).select{|i| i.gcd(n) == 1}.inject(:+)
end

def A000010(n)
  (1..n).map{|i| f(i)}
end

def A023896_1(n)
  (1..n).map{|i| g1(i)}
end

def A023896_2(n)
  (1..n).map{|i| g2(i)}
end

ary = A000010(69)
# OEIS A000010のデータ
ary0 =
[1,1,2,2,4,2,6,4,6,4,10,4,12,6,8,8,16,6,18,8,12,
 10,22,8,20,12,18,12,28,8,30,16,20,16,24,12,36,18,
 24,16,40,12,42,20,24,22,46,16,42,20,32,24,52,18,
 40,24,36,28,58,16,60,30,36,32,48,20,66,32,44]
# 一致の確認
p ary == ary0

ary1 = A023896_1(56)
ary2 = A023896_2(56)
# OEIS A023896のデータ
ary0 =
[1,1,3,4,10,6,21,16,27,20,55,24,78,42,60,64,136,
 54,171,80,126,110,253,96,250,156,243,168,406,120,
 465,256,330,272,420,216,666,342,468,320,820,252,
 903,440,540,506,1081,384,1029,500,816,624,1378,
 486,1100,672]
# 一致の確認
p ary1 == ary0
p ary2 == ary0

出力結果
true
true
true

160112

Ruby

単位分数の和が1 / 2(3)

p + q > n という条件をはずすとどうなるか調べてみた。

すなわち、

n を1 より大きい自然数とし、

互いに素となる正の整数p, q （ただし、0 < p < q ≦ n）で、

Σ(p, q) 1 / (pq)

を求めてみた。

def f(n)
  ary = []
  s = 0
  (2..n).each{|i|
    (1..i - 1).each{|j|
      s += 1r / (i * j) if i.gcd(j) == 1
    }
    ary << s
  }
  ary
end

p f(50)

出力結果
[(1/2), (1/1), (4/3), (7/4), (39/20), (23/10), (527/210), (463/168), (1469/504), (334/105), (3812/1155), (2803/792), (5723/1560), (229373/60060), (710887/180180), (66375/16016), (523689/123760), (2254127/510510), (131201039/29099070), (98005643/21162960), (1587557/335920), (15799073/3233230), (368539631/74364290), (232942821/45762640), (968698307/187210800), (70723666817/13385572200), (215198406259/40156716600), (534876161/97349616), (172178330879/31054527504), (943823276783/166363540200), (69160421412893/12033629407800), (16525794496429/2831442213600), (10032379128731/1698865328160), (1604617834569/267413986840), (622305953361/102851533400), (6691804682987/1085740999200), (583649404869259/93735639597600), (112299040878679/17809771523544), (339761515094597/53429314570632), (11511237187650241/1780977152354400), (1488661859008151/228903019581600), (16439677922891971/2489320337949900), (36411729017455/5468238801116), (4078482098102365/606736757410784), (2775287404425615383/409547311252279200), (5393065707566515283/784965679900201800), (1219770607484002541/176523382561289400), (619261627853593128913/88544128692742763040), (1454173851705467756797/206602966949733113760)]

160111(3)

Ruby


単位分数の和が1 / 2(2)

分母を互いに素な２つの数に分解してみた。

# -*- coding: cp932 -*-

def f(n)
  return [] if n < 2
  ary = [[1, 2]]
  a_ary = [[[1, 2]]]
  (3..n).each{|i|
    (1..i / 2).each{|j|
      if i.gcd(j) == 1
        ary.delete_at(ary.index([j, i - j]))
        ary << [j, i]
        ary << [i - j, i]
      end
    }
    a_ary << ary.clone.sort
  }
  a_ary
end

p f(1)
p f(2)
p ary = f(30)
puts '逆数の和が1 / 2になることを確認してみる'
p ary.map{|i| i.inject(0){|s, j| s += 1r / (j[0] * j[1])} == 1r / 2}

出力結果
[]
[[[1, 2]]]
[[[1, 2]], [[1, 3], [2, 3]], [[1, 4], [2, 3], [3, 4]], [[1, 5], [2, 5], [3, 4], [3, 5], [4, 5]], [[1, 6], [2, 5], [3, 4], [3, 5], [4, 5], [5, 6]], [[1, 7], [2, 7], [3, 5], [3, 7], [4, 5], [4, 7], [5, 6], [5, 7], [6, 7]], [[1, 8], [2, 7], [3, 7], [3, 8], [4, 5], [4, 7], [5, 6], [5, 7], [5, 8], [6, 7], [7, 8]], [[1, 9], [2, 9], [3, 7], [3, 8], [4, 7], [4, 9], [5, 6], [5, 7], [5, 8], [5, 9], [6, 7], [7, 8], [7, 9], [8, 9]], [[1, 10], [2, 9], [3, 8], [3, 10], [4, 7], [4, 9], [5, 6], [5, 7], [5, 8], [5, 9], [6, 7], [7, 8], [7, 9], [7, 10], [8, 9], [9, 10]], [[1, 11], [2, 11], [3, 10], [3, 11], [4, 9], [4, 11], [5, 7], [5, 8], [5, 9], [5, 11], [6, 7], [6, 11], [7, 8], [7, 9], [7, 10], [7, 11], [8, 9], [8, 11], [9, 10], [9, 11], [10, 11]], [[1, 12], [2, 11], [3, 10], [3, 11], [4, 9], [4, 11], [5, 8], [5, 9], [5, 11], [5, 12], [6, 7], [6, 11], [7, 8], [7, 9], [7, 10], [7, 11], [7, 12], [8, 9], [8, 11], [9, 10], [9, 11], [10, 11], [11, 12]], [[1, 13], [2, 13], [3, 11], [3, 13], [4, 11], [4, 13], [5, 9], [5, 11], [5, 12], [5, 13], [6, 11], [6, 13], [7, 8], [7, 9], [7, 10], [7, 11], [7, 12], [7, 13], [8, 9], [8, 11], [8, 13], [9, 10], [9, 11], [9, 13], [10, 11], [10, 13], [11, 12], [11, 13], [12, 13]], [[1, 14], [2, 13], [3, 13], [3, 14], [4, 11], [4, 13], [5, 11], [5, 12], [5, 13], [5, 14], [6, 11], [6, 13], [7, 8], [7, 9], [7, 10], [7, 11], [7, 12], [7, 13], [8, 9], [8, 11], [8, 13], [9, 10], [9, 11], [9, 13], [9, 14], [10, 11], [10, 13], [11, 12], [11, 13], [11, 14], [12, 13], [13, 14]], [[1, 15], [2, 15], [3, 13], [3, 14], [4, 13], [4, 15], [5, 11], [5, 12], [5, 13], [5, 14], [6, 11], [6, 13], [7, 9], [7, 10], [7, 11], [7, 12], [7, 13], [7, 15], [8, 9], [8, 11], [8, 13], [8, 15], [9, 10], [9, 11], [9, 13], [9, 14], [10, 11], [10, 13], [11, 12], [11, 13], [11, 14], [11, 15], [12, 13], [13, 14], [13, 15], [14, 15]], [[1, 16], [2, 15], [3, 14], [3, 16], [4, 13], [4, 15], [5, 12], [5, 13], [5, 14], [5, 16], [6, 11], [6, 13], [7, 10], [7, 11], [7, 12], [7, 13], [7, 15], [7, 16], [8, 9], [8, 11], [8, 13], [8, 15], [9, 10], [9, 11], [9, 13], [9, 14], [9, 16], [10, 11], [10, 13], [11, 12], [11, 13], [11, 14], [11, 15], [11, 16], [12, 13], [13, 14], [13, 15], [13, 16], [14, 15], [15, 16]], [[1, 17], [2, 17], [3, 16], [3, 17], [4, 15], [4, 17], [5, 13], [5, 14], [5, 16], [5, 17], [6, 13], [6, 17], [7, 11], [7, 12], [7, 13], [7, 15], [7, 16], [7, 17], [8, 11], [8, 13], [8, 15], [8, 17], [9, 10], [9, 11], [9, 13], [9, 14], [9, 16], [9, 17], [10, 11], [10, 13], [10, 17], [11, 12], [11, 13], [11, 14], [11, 15], [11, 16], [11, 17], [12, 13], [12, 17], [13, 14], [13, 15], [13, 16], [13, 17], [14, 15], [14, 17], [15, 16], [15, 17], [16, 17]], [[1, 18], [2, 17], [3, 16], [3, 17], [4, 15], [4, 17], [5, 14], [5, 16], [5, 17], [5, 18], [6, 13], [6, 17], [7, 12], [7, 13], [7, 15], [7, 16], [7, 17], [7, 18], [8, 11], [8, 13], [8, 15], [8, 17], [9, 10], [9, 11], [9, 13], [9, 14], [9, 16], [9, 17], [10, 11], [10, 13], [10, 17], [11, 12], [11, 13], [11, 14], [11, 15], [11, 16], [11, 17], [11, 18], [12, 13], [12, 17], [13, 14], [13, 15], [13, 16], [13, 17], [13, 18], [14, 15], [14, 17], [15, 16], [15, 17], [16, 17], [17, 18]], [[1, 19], [2, 19], [3, 17], [3, 19], [4, 17], [4, 19], [5, 16], [5, 17], [5, 18], [5, 19], [6, 17], [6, 19], [7, 13], [7, 15], [7, 16], [7, 17], [7, 18], [7, 19], [8, 13], [8, 15], [8, 17], [8, 19], [9, 11], [9, 13], [9, 14], [9, 16], [9, 17], [9, 19], [10, 11], [10, 13], [10, 17], [10, 19], [11, 12], [11, 13], [11, 14], [11, 15], [11, 16], [11, 17], [11, 18], [11, 19], [12, 13], [12, 17], [12, 19], [13, 14], [13, 15], [13, 16], [13, 17], [13, 18], [13, 19], [14, 15], [14, 17], [14, 19], [15, 16], [15, 17], [15, 19], [16, 17], [16, 19], [17, 18], [17, 19], [18, 19]], [[1, 20], [2, 19], [3, 19], [3, 20], [4, 17], [4, 19], [5, 16], [5, 17], [5, 18], [5, 19], [6, 17], [6, 19], [7, 15], [7, 16], [7, 17], [7, 18], [7, 19], [7, 20], [8, 13], [8, 15], [8, 17], [8, 19], [9, 13], [9, 14], [9, 16], [9, 17], [9, 19], [9, 20], [10, 11], [10, 13], [10, 17], [10, 19], [11, 12], [11, 13], [11, 14], [11, 15], [11, 16], [11, 17], [11, 18], [11, 19], [11, 20], [12, 13], [12, 17], [12, 19], [13, 14], [13, 15], [13, 16], [13, 17], [13, 18], [13, 19], [13, 20], [14, 15], [14, 17], [14, 19], [15, 16], [15, 17], [15, 19], [16, 17], [16, 19], [17, 18], [17, 19], [17, 20], [18, 19], [19, 20]], [[1, 21], [2, 21], [3, 19], [3, 20], [4, 19], [4, 21], [5, 17], [5, 18], [5, 19], [5, 21], [6, 17], [6, 19], [7, 15], [7, 16], [7, 17], [7, 18], [7, 19], [7, 20], [8, 15], [8, 17], [8, 19], [8, 21], [9, 13], [9, 14], [9, 16], [9, 17], [9, 19], [9, 20], [10, 13], [10, 17], [10, 19], [10, 21], [11, 12], [11, 13], [11, 14], [11, 15], [11, 16], [11, 17], [11, 18], [11, 19], [11, 20], [11, 21], [12, 13], [12, 17], [12, 19], [13, 14], [13, 15], [13, 16], [13, 17], [13, 18], [13, 19], [13, 20], [13, 21], [14, 15], [14, 17], [14, 19], [15, 16], [15, 17], [15, 19], [16, 17], [16, 19], [16, 21], [17, 18], [17, 19], [17, 20], [17, 21], [18, 19], [19, 20], [19, 21], [20, 21]], [[1, 22], [2, 21], [3, 20], [3, 22], [4, 19], [4, 21], [5, 18], [5, 19], [5, 21], [5, 22], [6, 17], [6, 19], [7, 16], [7, 17], [7, 18], [7, 19], [7, 20], [7, 22], [8, 15], [8, 17], [8, 19], [8, 21], [9, 14], [9, 16], [9, 17], [9, 19], [9, 20], [9, 22], [10, 13], [10, 17], [10, 19], [10, 21], [11, 12], [11, 13], [11, 14], [11, 15], [11, 16], [11, 17], [11, 18], [11, 19], [11, 20], [11, 21], [12, 13], [12, 17], [12, 19], [13, 14], [13, 15], [13, 16], [13, 17], [13, 18], [13, 19], [13, 20], [13, 21], [13, 22], [14, 15], [14, 17], [14, 19], [15, 16], [15, 17], [15, 19], [15, 22], [16, 17], [16, 19], [16, 21], [17, 18], [17, 19], [17, 20], [17, 21], [17, 22], [18, 19], [19, 20], [19, 21], [19, 22], [20, 21], [21, 22]], [[1, 23], [2, 23], [3, 22], [3, 23], [4, 21], [4, 23], [5, 19], [5, 21], [5, 22], [5, 23], [6, 19], [6, 23], [7, 17], [7, 18], [7, 19], [7, 20], [7, 22], [7, 23], [8, 17], [8, 19], [8, 21], [8, 23], [9, 16], [9, 17], [9, 19], [9, 20], [9, 22], [9, 23], [10, 17], [10, 19], [10, 21], [10, 23], [11, 13], [11, 14], [11, 15], [11, 16], [11, 17], [11, 18], [11, 19], [11, 20], [11, 21], [11, 23], [12, 13], [12, 17], [12, 19], [12, 23], [13, 14], [13, 15], [13, 16], [13, 17], [13, 18], [13, 19], [13, 20], [13, 21], [13, 22], [13, 23], [14, 15], [14, 17], [14, 19], [14, 23], [15, 16], [15, 17], [15, 19], [15, 22], [15, 23], [16, 17], [16, 19], [16, 21], [16, 23], [17, 18], [17, 19], [17, 20], [17, 21], [17, 22], [17, 23], [18, 19], [18, 23], [19, 20], [19, 21], [19, 22], [19, 23], [20, 21], [20, 23], [21, 22], [21, 23], [22, 23]], [[1, 24], [2, 23], [3, 22], [3, 23], [4, 21], [4, 23], [5, 21], [5, 22], [5, 23], [5, 24], [6, 19], [6, 23], [7, 18], [7, 19], [7, 20], [7, 22], [7, 23], [7, 24], [8, 17], [8, 19], [8, 21], [8, 23], [9, 16], [9, 17], [9, 19], [9, 20], [9, 22], [9, 23], [10, 17], [10, 19], [10, 21], [10, 23], [11, 14], [11, 15], [11, 16], [11, 17], [11, 18], [11, 19], [11, 20], [11, 21], [11, 23], [11, 24], [12, 13], [12, 17], [12, 19], [12, 23], [13, 14], [13, 15], [13, 16], [13, 17], [13, 18], [13, 19], [13, 20], [13, 21], [13, 22], [13, 23], [13, 24], [14, 15], [14, 17], [14, 19], [14, 23], [15, 16], [15, 17], [15, 19], [15, 22], [15, 23], [16, 17], [16, 19], [16, 21], [16, 23], [17, 18], [17, 19], [17, 20], [17, 21], [17, 22], [17, 23], [17, 24], [18, 19], [18, 23], [19, 20], [19, 21], [19, 22], [19, 23], [19, 24], [20, 21], [20, 23], [21, 22], [21, 23], [22, 23], [23, 24]], [[1, 25], [2, 25], [3, 23], [3, 25], [4, 23], [4, 25], [5, 21], [5, 22], [5, 23], [5, 24], [6, 23], [6, 25], [7, 19], [7, 20], [7, 22], [7, 23], [7, 24], [7, 25], [8, 19], [8, 21], [8, 23], [8, 25], [9, 17], [9, 19], [9, 20], [9, 22], [9, 23], [9, 25], [10, 17], [10, 19], [10, 21], [10, 23], [11, 15], [11, 16], [11, 17], [11, 18], [11, 19], [11, 20], [11, 21], [11, 23], [11, 24], [11, 25], [12, 17], [12, 19], [12, 23], [12, 25], [13, 14], [13, 15], [13, 16], [13, 17], [13, 18], [13, 19], [13, 20], [13, 21], [13, 22], [13, 23], [13, 24], [13, 25], [14, 15], [14, 17], [14, 19], [14, 23], [14, 25], [15, 16], [15, 17], [15, 19], [15, 22], [15, 23], [16, 17], [16, 19], [16, 21], [16, 23], [16, 25], [17, 18], [17, 19], [17, 20], [17, 21], [17, 22], [17, 23], [17, 24], [17, 25], [18, 19], [18, 23], [18, 25], [19, 20], [19, 21], [19, 22], [19, 23], [19, 24], [19, 25], [20, 21], [20, 23], [21, 22], [21, 23], [21, 25], [22, 23], [22, 25], [23, 24], [23, 25], [24, 25]], [[1, 26], [2, 25], [3, 25], [3, 26], [4, 23], [4, 25], [5, 22], [5, 23], [5, 24], [5, 26], [6, 23], [6, 25], [7, 20], [7, 22], [7, 23], [7, 24], [7, 25], [7, 26], [8, 19], [8, 21], [8, 23], [8, 25], [9, 19], [9, 20], [9, 22], [9, 23], [9, 25], [9, 26], [10, 17], [10, 19], [10, 21], [10, 23], [11, 16], [11, 17], [11, 18], [11, 19], [11, 20], [11, 21], [11, 23], [11, 24], [11, 25], [11, 26], [12, 17], [12, 19], [12, 23], [12, 25], [13, 14], [13, 15], [13, 16], [13, 17], [13, 18], [13, 19], [13, 20], [13, 21], [13, 22], [13, 23], [13, 24], [13, 25], [14, 15], [14, 17], [14, 19], [14, 23], [14, 25], [15, 16], [15, 17], [15, 19], [15, 22], [15, 23], [15, 26], [16, 17], [16, 19], [16, 21], [16, 23], [16, 25], [17, 18], [17, 19], [17, 20], [17, 21], [17, 22], [17, 23], [17, 24], [17, 25], [17, 26], [18, 19], [18, 23], [18, 25], [19, 20], [19, 21], [19, 22], [19, 23], [19, 24], [19, 25], [19, 26], [20, 21], [20, 23], [21, 22], [21, 23], [21, 25], [21, 26], [22, 23], [22, 25], [23, 24], [23, 25], [23, 26], [24, 25], [25, 26]], [[1, 27], [2, 27], [3, 25], [3, 26], [4, 25], [4, 27], [5, 23], [5, 24], [5, 26], [5, 27], [6, 23], [6, 25], [7, 22], [7, 23], [7, 24], [7, 25], [7, 26], [7, 27], [8, 21], [8, 23], [8, 25], [8, 27], [9, 19], [9, 20], [9, 22], [9, 23], [9, 25], [9, 26], [10, 19], [10, 21], [10, 23], [10, 27], [11, 17], [11, 18], [11, 19], [11, 20], [11, 21], [11, 23], [11, 24], [11, 25], [11, 26], [11, 27], [12, 17], [12, 19], [12, 23], [12, 25], [13, 15], [13, 16], [13, 17], [13, 18], [13, 19], [13, 20], [13, 21], [13, 22], [13, 23], [13, 24], [13, 25], [13, 27], [14, 15], [14, 17], [14, 19], [14, 23], [14, 25], [14, 27], [15, 16], [15, 17], [15, 19], [15, 22], [15, 23], [15, 26], [16, 17], [16, 19], [16, 21], [16, 23], [16, 25], [16, 27], [17, 18], [17, 19], [17, 20], [17, 21], [17, 22], [17, 23], [17, 24], [17, 25], [17, 26], [17, 27], [18, 19], [18, 23], [18, 25], [19, 20], [19, 21], [19, 22], [19, 23], [19, 24], [19, 25], [19, 26], [19, 27], [20, 21], [20, 23], [20, 27], [21, 22], [21, 23], [21, 25], [21, 26], [22, 23], [22, 25], [22, 27], [23, 24], [23, 25], [23, 26], [23, 27], [24, 25], [25, 26], [25, 27], [26, 27]], [[1, 28], [2, 27], [3, 26], [3, 28], [4, 25], [4, 27], [5, 24], [5, 26], [5, 27], [5, 28], [6, 23], [6, 25], [7, 22], [7, 23], [7, 24], [7, 25], [7, 26], [7, 27], [8, 21], [8, 23], [8, 25], [8, 27], [9, 20], [9, 22], [9, 23], [9, 25], [9, 26], [9, 28], [10, 19], [10, 21], [10, 23], [10, 27], [11, 18], [11, 19], [11, 20], [11, 21], [11, 23], [11, 24], [11, 25], [11, 26], [11, 27], [11, 28], [12, 17], [12, 19], [12, 23], [12, 25], [13, 16], [13, 17], [13, 18], [13, 19], [13, 20], [13, 21], [13, 22], [13, 23], [13, 24], [13, 25], [13, 27], [13, 28], [14, 15], [14, 17], [14, 19], [14, 23], [14, 25], [14, 27], [15, 16], [15, 17], [15, 19], [15, 22], [15, 23], [15, 26], [15, 28], [16, 17], [16, 19], [16, 21], [16, 23], [16, 25], [16, 27], [17, 18], [17, 19], [17, 20], [17, 21], [17, 22], [17, 23], [17, 24], [17, 25], [17, 26], [17, 27], [17, 28], [18, 19], [18, 23], [18, 25], [19, 20], [19, 21], [19, 22], [19, 23], [19, 24], [19, 25], [19, 26], [19, 27], [19, 28], [20, 21], [20, 23], [20, 27], [21, 22], [21, 23], [21, 25], [21, 26], [22, 23], [22, 25], [22, 27], [23, 24], [23, 25], [23, 26], [23, 27], [23, 28], [24, 25], [25, 26], [25, 27], [25, 28], [26, 27], [27, 28]], [[1, 29], [2, 29], [3, 28], [3, 29], [4, 27], [4, 29], [5, 26], [5, 27], [5, 28], [5, 29], [6, 25], [6, 29], [7, 23], [7, 24], [7, 25], [7, 26], [7, 27], [7, 29], [8, 23], [8, 25], [8, 27], [8, 29], [9, 22], [9, 23], [9, 25], [9, 26], [9, 28], [9, 29], [10, 21], [10, 23], [10, 27], [10, 29], [11, 19], [11, 20], [11, 21], [11, 23], [11, 24], [11, 25], [11, 26], [11, 27], [11, 28], [11, 29], [12, 19], [12, 23], [12, 25], [12, 29], [13, 17], [13, 18], [13, 19], [13, 20], [13, 21], [13, 22], [13, 23], [13, 24], [13, 25], [13, 27], [13, 28], [13, 29], [14, 17], [14, 19], [14, 23], [14, 25], [14, 27], [14, 29], [15, 16], [15, 17], [15, 19], [15, 22], [15, 23], [15, 26], [15, 28], [15, 29], [16, 17], [16, 19], [16, 21], [16, 23], [16, 25], [16, 27], [16, 29], [17, 18], [17, 19], [17, 20], [17, 21], [17, 22], [17, 23], [17, 24], [17, 25], [17, 26], [17, 27], [17, 28], [17, 29], [18, 19], [18, 23], [18, 25], [18, 29], [19, 20], [19, 21], [19, 22], [19, 23], [19, 24], [19, 25], [19, 26], [19, 27], [19, 28], [19, 29], [20, 21], [20, 23], [20, 27], [20, 29], [21, 22], [21, 23], [21, 25], [21, 26], [21, 29], [22, 23], [22, 25], [22, 27], [22, 29], [23, 24], [23, 25], [23, 26], [23, 27], [23, 28], [23, 29], [24, 25], [24, 29], [25, 26], [25, 27], [25, 28], [25, 29], [26, 27], [26, 29], [27, 28], [27, 29], [28, 29]], [[1, 30], [2, 29], [3, 28], [3, 29], [4, 27], [4, 29], [5, 26], [5, 27], [5, 28], [5, 29], [6, 25], [6, 29], [7, 24], [7, 25], [7, 26], [7, 27], [7, 29], [7, 30], [8, 23], [8, 25], [8, 27], [8, 29], [9, 22], [9, 23], [9, 25], [9, 26], [9, 28], [9, 29], [10, 21], [10, 23], [10, 27], [10, 29], [11, 20], [11, 21], [11, 23], [11, 24], [11, 25], [11, 26], [11, 27], [11, 28], [11, 29], [11, 30], [12, 19], [12, 23], [12, 25], [12, 29], [13, 18], [13, 19], [13, 20], [13, 21], [13, 22], [13, 23], [13, 24], [13, 25], [13, 27], [13, 28], [13, 29], [13, 30], [14, 17], [14, 19], [14, 23], [14, 25], [14, 27], [14, 29], [15, 16], [15, 17], [15, 19], [15, 22], [15, 23], [15, 26], [15, 28], [15, 29], [16, 17], [16, 19], [16, 21], [16, 23], [16, 25], [16, 27], [16, 29], [17, 18], [17, 19], [17, 20], [17, 21], [17, 22], [17, 23], [17, 24], [17, 25], [17, 26], [17, 27], [17, 28], [17, 29], [17, 30], [18, 19], [18, 23], [18, 25], [18, 29], [19, 20], [19, 21], [19, 22], [19, 23], [19, 24], [19, 25], [19, 26], [19, 27], [19, 28], [19, 29], [19, 30], [20, 21], [20, 23], [20, 27], [20, 29], [21, 22], [21, 23], [21, 25], [21, 26], [21, 29], [22, 23], [22, 25], [22, 27], [22, 29], [23, 24], [23, 25], [23, 26], [23, 27], [23, 28], [23, 29], [23, 30], [24, 25], [24, 29], [25, 26], [25, 27], [25, 28], [25, 29], [26, 27], [26, 29], [27, 28], [27, 29], [28, 29], [29, 30]]]
逆数の和が1 / 2になることを確認してみる
[true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true]

160111(2)

Ruby


単位分数の和が1 / 2(1)

http://www004.upp.so-net.ne.jp/s_honma/mathbun/mathbun339.htm
を読んで、分母のみを出力してみた。

# -*- coding: cp932 -*-

def f(n)
  return [] if n < 2
  ary = [2]
  a_ary = [[2]]
  (3..n).each{|i|
    (1..i / 2).each{|j|
      if i.gcd(j) == 1
        ary.delete_at(ary.index(j * (i - j)))
        ary << i * j
        ary << i * (i - j)
      end
    }
    a_ary << ary.clone.sort
  }
  a_ary
end

p f(1)
p f(2)
p ary = f(30)
puts '逆数の和が1 / 2になることを確認してみる'
p ary.map{|i| i.inject(0){|s, j| s += 1r / j} == 1r / 2}

出力結果
[]
[[2]]
[[2], [3, 6], [4, 6, 12], [5, 10, 12, 15, 20], [6, 10, 12, 15, 20, 30], [7, 14, 15, 20, 21, 28, 30, 35, 42], [8, 14, 20, 21, 24, 28, 30, 35, 40, 42, 56], [9, 18, 21, 24, 28, 30, 35, 36, 40, 42, 45, 56, 63, 72], [10, 18, 24, 28, 30, 30, 35, 36, 40, 42, 45, 56, 63, 70, 72, 90], [11, 22, 30, 33, 35, 36, 40, 42, 44, 45, 55, 56, 63, 66, 70, 72, 77, 88, 90, 99, 110], [12, 22, 30, 33, 36, 40, 42, 44, 45, 55, 56, 60, 63, 66, 70, 72, 77, 84, 88, 90, 99, 110, 132], [13, 26, 33, 39, 44, 45, 52, 55, 56, 60, 63, 65, 66, 70, 72, 77, 78, 84, 88, 90, 91, 99, 104, 110, 117, 130, 132, 143, 156], [14, 26, 39, 42, 44, 52, 55, 56, 60, 63, 65, 66, 70, 70, 72, 77, 78, 84, 88, 90, 91, 99, 104, 110, 117, 126, 130, 132, 143, 154, 156, 182], [15, 30, 39, 42, 52, 55, 60, 60, 63, 65, 66, 70, 70, 72, 77, 78, 84, 88, 90, 91, 99, 104, 105, 110, 117, 120, 126, 130, 132, 143, 154, 156, 165, 182, 195, 210], [16, 30, 42, 48, 52, 60, 60, 65, 66, 70, 70, 72, 77, 78, 80, 84, 88, 90, 91, 99, 104, 105, 110, 112, 117, 120, 126, 130, 132, 143, 144, 154, 156, 165, 176, 182, 195, 208, 210, 240], [17, 34, 48, 51, 60, 65, 68, 70, 77, 78, 80, 84, 85, 88, 90, 91, 99, 102, 104, 105, 110, 112, 117, 119, 120, 126, 130, 132, 136, 143, 144, 153, 154, 156, 165, 170, 176, 182, 187, 195, 204, 208, 210, 221, 238, 240, 255, 272], [18, 34, 48, 51, 60, 68, 70, 78, 80, 84, 85, 88, 90, 90, 91, 99, 102, 104, 105, 110, 112, 117, 119, 120, 126, 126, 130, 132, 136, 143, 144, 153, 154, 156, 165, 170, 176, 182, 187, 195, 198, 204, 208, 210, 221, 234, 238, 240, 255, 272, 306], [19, 38, 51, 57, 68, 76, 80, 85, 90, 91, 95, 99, 102, 104, 105, 110, 112, 114, 117, 119, 120, 126, 126, 130, 132, 133, 136, 143, 144, 152, 153, 154, 156, 165, 170, 171, 176, 182, 187, 190, 195, 198, 204, 208, 209, 210, 221, 228, 234, 238, 240, 247, 255, 266, 272, 285, 304, 306, 323, 342], [20, 38, 57, 60, 68, 76, 80, 85, 90, 95, 102, 104, 105, 110, 112, 114, 117, 119, 120, 126, 126, 130, 132, 133, 136, 140, 143, 144, 152, 153, 154, 156, 165, 170, 171, 176, 180, 182, 187, 190, 195, 198, 204, 208, 209, 210, 220, 221, 228, 234, 238, 240, 247, 255, 260, 266, 272, 285, 304, 306, 323, 340, 342, 380], [21, 42, 57, 60, 76, 84, 85, 90, 95, 102, 105, 105, 112, 114, 117, 119, 120, 126, 126, 130, 132, 133, 136, 140, 143, 144, 152, 153, 154, 156, 165, 168, 170, 171, 176, 180, 182, 187, 190, 195, 198, 204, 208, 209, 210, 210, 220, 221, 228, 231, 234, 238, 240, 247, 255, 260, 266, 272, 273, 285, 304, 306, 323, 336, 340, 342, 357, 380, 399, 420], [22, 42, 60, 66, 76, 84, 90, 95, 102, 105, 110, 112, 114, 119, 120, 126, 126, 130, 132, 133, 136, 140, 143, 144, 152, 153, 154, 154, 156, 165, 168, 170, 171, 176, 180, 182, 187, 190, 195, 198, 198, 204, 208, 209, 210, 210, 220, 221, 228, 231, 234, 238, 240, 247, 255, 260, 266, 272, 273, 285, 286, 304, 306, 323, 330, 336, 340, 342, 357, 374, 380, 399, 418, 420, 462], [23, 46, 66, 69, 84, 92, 95, 105, 110, 114, 115, 119, 126, 133, 136, 138, 140, 143, 144, 152, 153, 154, 154, 156, 161, 165, 168, 170, 171, 176, 180, 182, 184, 187, 190, 195, 198, 198, 204, 207, 208, 209, 210, 210, 220, 221, 228, 230, 231, 234, 238, 240, 247, 253, 255, 260, 266, 272, 273, 276, 285, 286, 299, 304, 306, 322, 323, 330, 336, 340, 342, 345, 357, 368, 374, 380, 391, 399, 414, 418, 420, 437, 460, 462, 483, 506], [24, 46, 66, 69, 84, 92, 105, 110, 114, 115, 120, 126, 133, 136, 138, 140, 144, 152, 153, 154, 154, 156, 161, 165, 168, 168, 170, 171, 176, 180, 182, 184, 187, 190, 195, 198, 198, 204, 207, 208, 209, 210, 210, 220, 221, 228, 230, 231, 234, 238, 240, 247, 253, 255, 260, 264, 266, 272, 273, 276, 285, 286, 299, 304, 306, 312, 322, 323, 330, 336, 340, 342, 345, 357, 368, 374, 380, 391, 399, 408, 414, 418, 420, 437, 456, 460, 462, 483, 506, 552], [25, 50, 69, 75, 92, 100, 105, 110, 115, 120, 133, 138, 140, 150, 152, 153, 154, 161, 165, 168, 168, 170, 171, 175, 176, 180, 182, 184, 187, 190, 195, 198, 198, 200, 204, 207, 208, 209, 210, 210, 220, 221, 225, 228, 230, 231, 234, 238, 240, 247, 253, 255, 260, 264, 266, 272, 273, 275, 276, 285, 286, 299, 300, 304, 306, 312, 322, 323, 325, 330, 336, 340, 342, 345, 350, 357, 368, 374, 380, 391, 399, 400, 408, 414, 418, 420, 425, 437, 450, 456, 460, 462, 475, 483, 506, 525, 550, 552, 575, 600], [26, 50, 75, 78, 92, 100, 110, 115, 120, 130, 138, 140, 150, 152, 154, 161, 168, 168, 170, 171, 175, 176, 180, 182, 182, 184, 187, 190, 195, 198, 198, 200, 204, 207, 208, 209, 210, 210, 220, 221, 225, 228, 230, 231, 234, 234, 238, 240, 247, 253, 255, 260, 264, 266, 272, 273, 275, 276, 285, 286, 286, 299, 300, 304, 306, 312, 322, 323, 325, 330, 336, 340, 342, 345, 350, 357, 368, 374, 380, 390, 391, 399, 400, 408, 414, 418, 420, 425, 437, 442, 450, 456, 460, 462, 475, 483, 494, 506, 525, 546, 550, 552, 575, 598, 600, 650], [27, 54, 75, 78, 100, 108, 115, 120, 130, 135, 138, 150, 154, 161, 168, 168, 171, 175, 180, 182, 184, 187, 189, 190, 195, 198, 198, 200, 204, 207, 208, 209, 210, 210, 216, 220, 221, 225, 228, 230, 231, 234, 234, 238, 240, 247, 253, 255, 260, 264, 266, 270, 272, 273, 275, 276, 285, 286, 286, 297, 299, 300, 304, 306, 312, 322, 323, 325, 330, 336, 340, 342, 345, 350, 351, 357, 368, 374, 378, 380, 390, 391, 399, 400, 408, 414, 418, 420, 425, 432, 437, 442, 450, 456, 459, 460, 462, 475, 483, 494, 506, 513, 525, 540, 546, 550, 552, 575, 594, 598, 600, 621, 650, 675, 702], [28, 54, 78, 84, 100, 108, 120, 130, 135, 138, 140, 150, 154, 161, 168, 168, 175, 180, 182, 184, 189, 190, 198, 198, 200, 204, 207, 208, 209, 210, 210, 216, 220, 221, 225, 228, 230, 231, 234, 234, 238, 240, 247, 252, 253, 255, 260, 264, 266, 270, 272, 273, 275, 276, 285, 286, 286, 297, 299, 300, 304, 306, 308, 312, 322, 323, 325, 330, 336, 340, 342, 345, 350, 351, 357, 364, 368, 374, 378, 380, 390, 391, 399, 400, 408, 414, 418, 420, 420, 425, 432, 437, 442, 450, 456, 459, 460, 462, 475, 476, 483, 494, 506, 513, 525, 532, 540, 546, 550, 552, 575, 594, 598, 600, 621, 644, 650, 675, 700, 702, 756], [29, 58, 84, 87, 108, 116, 130, 135, 140, 145, 150, 161, 168, 174, 175, 182, 184, 189, 198, 200, 203, 207, 209, 210, 216, 220, 221, 225, 228, 230, 231, 232, 234, 234, 238, 240, 247, 252, 253, 255, 260, 261, 264, 266, 270, 272, 273, 275, 276, 285, 286, 286, 290, 297, 299, 300, 304, 306, 308, 312, 319, 322, 323, 325, 330, 336, 340, 342, 345, 348, 350, 351, 357, 364, 368, 374, 377, 378, 380, 390, 391, 399, 400, 406, 408, 414, 418, 420, 420, 425, 432, 435, 437, 442, 450, 456, 459, 460, 462, 464, 475, 476, 483, 493, 494, 506, 513, 522, 525, 532, 540, 546, 550, 551, 552, 575, 580, 594, 598, 600, 609, 621, 638, 644, 650, 667, 675, 696, 700, 702, 725, 754, 756, 783, 812], [30, 58, 84, 87, 108, 116, 130, 135, 140, 145, 150, 168, 174, 175, 182, 184, 189, 198, 200, 203, 207, 210, 210, 216, 220, 225, 228, 230, 231, 232, 234, 234, 238, 240, 247, 252, 253, 255, 260, 261, 264, 266, 270, 272, 273, 275, 276, 285, 286, 286, 290, 297, 299, 300, 304, 306, 308, 312, 319, 322, 323, 325, 330, 330, 336, 340, 342, 345, 348, 350, 351, 357, 364, 368, 374, 377, 378, 380, 390, 390, 391, 399, 400, 406, 408, 414, 418, 420, 420, 425, 432, 435, 437, 442, 450, 456, 459, 460, 462, 464, 475, 476, 483, 493, 494, 506, 510, 513, 522, 525, 532, 540, 546, 550, 551, 552, 570, 575, 580, 594, 598, 600, 609, 621, 638, 644, 650, 667, 675, 690, 696, 700, 702, 725, 754, 756, 783, 812, 870]]
逆数の和が1 / 2になることを確認してみる
[true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true]

160111

Ruby


逆数をとり、単位分数を加えていく数列

以下のような操作を繰り返してみた。
0 + 1 / 2 = 1 / 2
2 / 1 + 1 / 3 = 7 / 3
3 / 7 + 1 / 4 = 19 / 28
28 / 19 + 1 / 5 = 159 / 95
…

def f(k, n)
  s = 0
  ary = []
  (k..k + n - 1).each{|i|
    s += 1r / i
    ary << s
    s = 1r / s
  }
  ary
end

p f(1, 50)
p f(2, 50)
p f(3, 50)
p f(10, 50)
p f(100, 50)

出力結果
[(1/1), (3/2), (1/1), (5/4), (1/1), (7/6), (1/1), (9/8), (1/1), (11/10), (1/1), (13/12), (1/1), (15/14), (1/1), (17/16), (1/1), (19/18), (1/1), (21/20), (1/1), (23/22), (1/1), (25/24), (1/1), (27/26), (1/1), (29/28), (1/1), (31/30), (1/1), (33/32), (1/1), (35/34), (1/1), (37/36), (1/1), (39/38), (1/1), (41/40), (1/1), (43/42), (1/1), (45/44), (1/1), (47/46), (1/1), (49/48), (1/1), (51/50)]
[(1/2), (7/3), (19/28), (159/95), (81/106), (823/567), (5359/6584), (64615/48231), (21877/25846), (306183/240647), (354883/408244), (5662055/4613479), (1433689/1617730), (25699639/21505335), (369784999/411194224), (7360086807/6286344983), (1487830821/1635574846), (32563752895/28268785599), (23917578595/26051002316), (570988627231/502269150495), (96040578001/103816114042), (2483811200967/2208933294023), (6166467806391/6623496535912), (171753881204191/154161695159775), (24733479025789/26423674031414), (738172677873967/667803933696303), (396666996354499/421812958785124), (12629242801123095/11503342894280471), (1589909020575721/1683899040149746), (53790779265217847/49287179637847351), (1630980527676333079/1721304936486971104), (58434043431746379511/53822357413318991607), (6534235970534920741/6874593344911338766), (247145003042431777551/228698258968722225935), (104694843529832492731/109842223574414123356), (4168857115783155056903/3873709210603802231047), (419306944927223379049/438827064819279479674), (17533562472879123086335/16352970852161711782911), (26866095862373903776111/28053699956606596938136), (1177067794083244378239687/1101509930357330054820551), (107574795621521783856469/112101694674594702689494), (4927947666629093999504711/4625716211725436705828167), (1722805462665688504594579/1791980969683306908910804), (82361949098414499405580759/77526245819955982706756055), (6897106345588638381694441/7161908617253434730920066), (343506811356500070734937543/324163998242666003939638727), (1767042080778274251093066271/1832036327234667043919666896), (91536822115276959403156744175/86585061958135438303560247279), (7073263872035278199329870573/7322945769222156752252539534), (380543498102365272564209386807/360736457473799188165823399223)]
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[(1/10), (111/11), (27/148), (1951/351), (6865/27314), (16663/4119), (82567/266608), (4614903/1403639), (3320045/9229806), (178686359/63080855), (1440303459/3573727180), (173443479/68585879), (13903577/31535178), (739212671/319782271), (8413987175/17741104104), (18077663591/8413987175), (1401427989/2781179014), (8499251263/4204283967), (126219202339/237979035364), (7027611227895/3660356867831), (519281410057/937014830386), (29566741152023/16097723711767), (544693899928567/946135716864736), (31767172556464855/17974898697642711), (2224614976734661/3737314418407630), (2714910604510239/1589010697667615), (739744391611659/1206626935337884), (45384940999113367/27370542489631383), (3006829793919961/4777362210432986), (189323956000806415/117266361962878479), (195199137380637823/302918329601290264), (12614850651033538647/8003164632606150743), (38749751691165763317/58869303038156513686), (2570129782331895851815/1666239322720127822631), (627145950264607603699/934592648120689400660), (42683815115695630633399/28221567761907342166455), (2534737111840138696201/3711636097017011359426), (176981633671639672589223/119132644256486518721447), (655038728664776952357631/943902046248744920475856), (957270183568434246034175/655038728664776952357631), (1348368264672291274556629/1914540367136868492068350), (98989926988652584370042479/68766781498286855002388079), (21744748904731177778072323/30458439073431564421551532), (1636042019796604092120303519/1152471691950752422237833119), (788512510927620183863744345/1090694679864402728080202346), (2431068796138790809130994935/1734727524040764404500237559), (2032159390661665254309067311/2778364338444332353292565640), (17822102964665401043553923199/12870342807523879943957426303), (908801409989358427815796253/1229110549287269037486477462), (73426323817938231639517966511/53619283189372147241131978927)]
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160109(6)

Ruby


9 × (10^(18n) - 1) / 19 の性質について(5)

k × (10^((c - 1)n) - m) / c で
2, … , c - 1 倍すると大体数字がずれた状態になる
ものを選んでみた。

def f(k, n, c)
  m = (k * 10 ** ((c - 1) * n)) % c
  return (k * 10 ** ((c - 1) * n) - m) / c
end

n = 2
# 7は除きました。また、出力結果が多くなるので97以降も除きました。
c_ary = [17, 19, 23, 29, 47, 59, 61]
ary = []
c_ary.each{|c|
  ary += (0..n).inject([]){|s, i| s + (1..c - 1).map{|k| f(k, i, c)}}.uniq!.sort
}
p ary.uniq!.sort

出力結果
[0, 588235294117647, 1176470588235294, 1764705882352941, 2352941176470588, 2941176470588235, 3529411764705882, 4117647058823529, 4705882352941176, 5294117647058823, 5882352941176470, 6470588235294117, 7058823529411764, 7647058823529411, 8235294117647058, 8823529411764705, 9411764705882352, 52631578947368421, 105263157894736842, 157894736842105263, 210526315789473684, 263157894736842105, 315789473684210526, 368421052631578947, 421052631578947368, 473684210526315789, 526315789473684210, 578947368421052631, 631578947368421052, 684210526315789473, 736842105263157894, 789473684210526315, 842105263157894736, 894736842105263157, 947368421052631578, 434782608695652173913, 869565217391304347826, 1304347826086956521739, 1739130434782608695652, 2173913043478260869565, 2608695652173913043478, 3043478260869565217391, 3478260869565217391304, 3913043478260869565217, 4347826086956521739130, 4782608695652173913043, 5217391304347826086956, 5652173913043478260869, 6086956521739130434782, 6521739130434782608695, 6956521739130434782608, 7391304347826086956521, 7826086956521739130434, 8260869565217391304347, 8695652173913043478260, 9130434782608695652173, 9565217391304347826086, 344827586206896551724137931, 689655172413793103448275862, 1034482758620689655172413793, 1379310344827586206896551724, 1724137931034482758620689655, 2068965517241379310344827586, 2413793103448275862068965517, 2758620689655172413793103448, 3103448275862068965517241379, 3448275862068965517241379310, 3793103448275862068965517241, 4137931034482758620689655172, 4482758620689655172413793103, 4827586206896551724137931034, 5172413793103448275862068965, 5517241379310344827586206896, 5862068965517241379310344827, 6206896551724137931034482758, 6551724137931034482758620689, 6896551724137931034482758620, 7241379310344827586206896551, 7586206896551724137931034482, 7931034482758620689655172413, 8275862068965517241379310344, 8620689655172413793103448275, 8965517241379310344827586206, 9310344827586206896551724137, 9655172413793103448275862068, 5882352941176470588235294117647, 11764705882352941176470588235294, 17647058823529411764705882352941, 23529411764705882352941176470588, 29411764705882352941176470588235, 35294117647058823529411764705882, 41176470588235294117647058823529, 47058823529411764705882352941176, 52941176470588235294117647058823, 58823529411764705882352941176470, 64705882352941176470588235294117, 70588235294117647058823529411764, 76470588235294117647058823529411, 82352941176470588235294117647058, 88235294117647058823529411764705, 94117647058823529411764705882352, 52631578947368421052631578947368421, 105263157894736842105263157894736842, 157894736842105263157894736842105263, 210526315789473684210526315789473684, 263157894736842105263157894736842105, 315789473684210526315789473684210526, 368421052631578947368421052631578947, 421052631578947368421052631578947368, 473684210526315789473684210526315789, 526315789473684210526315789473684210, 578947368421052631578947368421052631, 631578947368421052631578947368421052, 684210526315789473684210526315789473, 736842105263157894736842105263157894, 789473684210526315789473684210526315, 842105263157894736842105263157894736, 894736842105263157894736842105263157, 947368421052631578947368421052631578, 4347826086956521739130434782608695652173913, 8695652173913043478260869565217391304347826, 13043478260869565217391304347826086956521739, 17391304347826086956521739130434782608695652, 21739130434782608695652173913043478260869565, 26086956521739130434782608695652173913043478, 30434782608695652173913043478260869565217391, 34782608695652173913043478260869565217391304, 39130434782608695652173913043478260869565217, 43478260869565217391304347826086956521739130, 47826086956521739130434782608695652173913043, 52173913043478260869565217391304347826086956, 56521739130434782608695652173913043478260869, 60869565217391304347826086956521739130434782, 65217391304347826086956521739130434782608695, 69565217391304347826086956521739130434782608, 73913043478260869565217391304347826086956521, 78260869565217391304347826086956521739130434, 82608695652173913043478260869565217391304347, 86956521739130434782608695652173913043478260, 91304347826086956521739130434782608695652173, 95652173913043478260869565217391304347826086, 212765957446808510638297872340425531914893617, 425531914893617021276595744680851063829787234, 638297872340425531914893617021276595744680851, 851063829787234042553191489361702127659574468, 1063829787234042553191489361702127659574468085, 1276595744680851063829787234042553191489361702, 1489361702127659574468085106382978723404255319, 1702127659574468085106382978723404255319148936, 1914893617021276595744680851063829787234042553, 2127659574468085106382978723404255319148936170, 2340425531914893617021276595744680851063829787, 2553191489361702127659574468085106382978723404, 2765957446808510638297872340425531914893617021, 2978723404255319148936170212765957446808510638, 3191489361702127659574468085106382978723404255, 3404255319148936170212765957446808510638297872, 3617021276595744680851063829787234042553191489, 3829787234042553191489361702127659574468085106, 4042553191489361702127659574468085106382978723, 4255319148936170212765957446808510638297872340, 4468085106382978723404255319148936170212765957, 4680851063829787234042553191489361702127659574, 4893617021276595744680851063829787234042553191, 5106382978723404255319148936170212765957446808, 5319148936170212765957446808510638297872340425, 5531914893617021276595744680851063829787234042, 5744680851063829787234042553191489361702127659, 5957446808510638297872340425531914893617021276, 6170212765957446808510638297872340425531914893, 6382978723404255319148936170212765957446808510, 6595744680851063829787234042553191489361702127, 6808510638297872340425531914893617021276595744, 7021276595744680851063829787234042553191489361, 7234042553191489361702127659574468085106382978, 7446808510638297872340425531914893617021276595, 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79661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067, 81355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728, 83050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389, 84745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050, 86440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711, 88135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372, 89830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033, 91525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694, 93220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355, 94915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016, 96610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677, 98305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338, 16393442622950819672131147540983606557377049180327868852459016393442622950819672131147540983606557377049180327868852459, 32786885245901639344262295081967213114754098360655737704918032786885245901639344262295081967213114754098360655737704918, 49180327868852459016393442622950819672131147540983606557377049180327868852459016393442622950819672131147540983606557377, 65573770491803278688524590163934426229508196721311475409836065573770491803278688524590163934426229508196721311475409836, 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213114754098360655737704918032786885245901639344262295081967213114754098360655737704918032786885245901639344262295081967, 229508196721311475409836065573770491803278688524590163934426229508196721311475409836065573770491803278688524590163934426, 245901639344262295081967213114754098360655737704918032786885245901639344262295081967213114754098360655737704918032786885, 262295081967213114754098360655737704918032786885245901639344262295081967213114754098360655737704918032786885245901639344, 278688524590163934426229508196721311475409836065573770491803278688524590163934426229508196721311475409836065573770491803, 295081967213114754098360655737704918032786885245901639344262295081967213114754098360655737704918032786885245901639344262, 311475409836065573770491803278688524590163934426229508196721311475409836065573770491803278688524590163934426229508196721, 327868852459016393442622950819672131147540983606557377049180327868852459016393442622950819672131147540983606557377049180, 344262295081967213114754098360655737704918032786885245901639344262295081967213114754098360655737704918032786885245901639, 360655737704918032786885245901639344262295081967213114754098360655737704918032786885245901639344262295081967213114754098, 377049180327868852459016393442622950819672131147540983606557377049180327868852459016393442622950819672131147540983606557, 393442622950819672131147540983606557377049180327868852459016393442622950819672131147540983606557377049180327868852459016, 409836065573770491803278688524590163934426229508196721311475409836065573770491803278688524590163934426229508196721311475, 426229508196721311475409836065573770491803278688524590163934426229508196721311475409836065573770491803278688524590163934, 442622950819672131147540983606557377049180327868852459016393442622950819672131147540983606557377049180327868852459016393, 459016393442622950819672131147540983606557377049180327868852459016393442622950819672131147540983606557377049180327868852, 475409836065573770491803278688524590163934426229508196721311475409836065573770491803278688524590163934426229508196721311, 491803278688524590163934426229508196721311475409836065573770491803278688524590163934426229508196721311475409836065573770, 508196721311475409836065573770491803278688524590163934426229508196721311475409836065573770491803278688524590163934426229, 524590163934426229508196721311475409836065573770491803278688524590163934426229508196721311475409836065573770491803278688, 540983606557377049180327868852459016393442622950819672131147540983606557377049180327868852459016393442622950819672131147, 557377049180327868852459016393442622950819672131147540983606557377049180327868852459016393442622950819672131147540983606, 573770491803278688524590163934426229508196721311475409836065573770491803278688524590163934426229508196721311475409836065, 590163934426229508196721311475409836065573770491803278688524590163934426229508196721311475409836065573770491803278688524, 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868852459016393442622950819672131147540983606557377049180327868852459016393442622950819672131147540983606557377049180327, 885245901639344262295081967213114754098360655737704918032786885245901639344262295081967213114754098360655737704918032786, 901639344262295081967213114754098360655737704918032786885245901639344262295081967213114754098360655737704918032786885245, 918032786885245901639344262295081967213114754098360655737704918032786885245901639344262295081967213114754098360655737704, 934426229508196721311475409836065573770491803278688524590163934426229508196721311475409836065573770491803278688524590163, 950819672131147540983606557377049180327868852459016393442622950819672131147540983606557377049180327868852459016393442622, 967213114754098360655737704918032786885245901639344262295081967213114754098360655737704918032786885245901639344262295081, 983606557377049180327868852459016393442622950819672131147540983606557377049180327868852459016393442622950819672131147540]

160109(5)

Ruby


9 × (10^(18n) - 1) / 19 の性質について(4)

分母が19 でない場合についても考える。

k × (10^(16n) - m) / 17 で
2, … , 16 倍すると大体数字がずれた状態になる
ものを選んでみた。
例えば、
4705882352941176 × 1 = 4705882352941176
4705882352941176 × 2 = 9411764705882352
4705882352941176 × 3 = 14117647058823528
4705882352941176 × 4 = 18823529411764704
4705882352941176 × 5 = 23529411764705880
4705882352941176 × 6 = 28235294117647056
4705882352941176 × 7 = 32941176470588232
4705882352941176 × 8 = 37647058823529408
4705882352941176 × 9 = 42352941176470584
4705882352941176 × 10 = 47058823529411760
4705882352941176 × 11 = 51764705882352936
4705882352941176 × 12 = 56470588235294112
4705882352941176 × 13 = 61176470588235288
4705882352941176 × 14 = 65882352941176464
4705882352941176 × 15 = 70588235294117640
4705882352941176 × 16 = 75294117647058816

def f(k, n, c)
  m = (k * 10 ** ((c - 1) * n)) % c
  return (k * 10 ** ((c - 1) * n) - m) / c
end

n = 3
p ary = (0..n).inject([]){|s, i| s + (1..16).map{|k| f(k, i, 17)}}.uniq!.sort
# それぞれj倍したもの
ary.each{|i| p (1..16).map{|j| i * j}}

出力結果
[0, 588235294117647, 1176470588235294, 1764705882352941, 2352941176470588, 2941176470588235, 3529411764705882, 4117647058823529, 4705882352941176, 5294117647058823, 5882352941176470, 6470588235294117, 7058823529411764, 7647058823529411, 8235294117647058, 8823529411764705, 9411764705882352, 5882352941176470588235294117647, 11764705882352941176470588235294, 17647058823529411764705882352941, 23529411764705882352941176470588, 29411764705882352941176470588235, 35294117647058823529411764705882, 41176470588235294117647058823529, 47058823529411764705882352941176, 52941176470588235294117647058823, 58823529411764705882352941176470, 64705882352941176470588235294117, 70588235294117647058823529411764, 76470588235294117647058823529411, 82352941176470588235294117647058, 88235294117647058823529411764705, 94117647058823529411764705882352, 58823529411764705882352941176470588235294117647, 117647058823529411764705882352941176470588235294, 176470588235294117647058823529411764705882352941, 235294117647058823529411764705882352941176470588, 294117647058823529411764705882352941176470588235, 352941176470588235294117647058823529411764705882, 411764705882352941176470588235294117647058823529, 470588235294117647058823529411764705882352941176, 529411764705882352941176470588235294117647058823, 588235294117647058823529411764705882352941176470, 647058823529411764705882352941176470588235294117, 705882352941176470588235294117647058823529411764, 764705882352941176470588235294117647058823529411, 823529411764705882352941176470588235294117647058, 882352941176470588235294117647058823529411764705, 941176470588235294117647058823529411764705882352]
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
[588235294117647, 1176470588235294, 1764705882352941, 2352941176470588, 2941176470588235, 3529411764705882, 4117647058823529, 4705882352941176, 5294117647058823, 5882352941176470, 6470588235294117, 7058823529411764, 7647058823529411, 8235294117647058, 8823529411764705, 9411764705882352]
[1176470588235294, 2352941176470588, 3529411764705882, 4705882352941176, 5882352941176470, 7058823529411764, 8235294117647058, 9411764705882352, 10588235294117646, 11764705882352940, 12941176470588234, 14117647058823528, 15294117647058822, 16470588235294116, 17647058823529410, 18823529411764704]
[1764705882352941, 3529411764705882, 5294117647058823, 7058823529411764, 8823529411764705, 10588235294117646, 12352941176470587, 14117647058823528, 15882352941176469, 17647058823529410, 19411764705882351, 21176470588235292, 22941176470588233, 24705882352941174, 26470588235294115, 28235294117647056]
[2352941176470588, 4705882352941176, 7058823529411764, 9411764705882352, 11764705882352940, 14117647058823528, 16470588235294116, 18823529411764704, 21176470588235292, 23529411764705880, 25882352941176468, 28235294117647056, 30588235294117644, 32941176470588232, 35294117647058820, 37647058823529408]
[2941176470588235, 5882352941176470, 8823529411764705, 11764705882352940, 14705882352941175, 17647058823529410, 20588235294117645, 23529411764705880, 26470588235294115, 29411764705882350, 32352941176470585, 35294117647058820, 38235294117647055, 41176470588235290, 44117647058823525, 47058823529411760]
[3529411764705882, 7058823529411764, 10588235294117646, 14117647058823528, 17647058823529410, 21176470588235292, 24705882352941174, 28235294117647056, 31764705882352938, 35294117647058820, 38823529411764702, 42352941176470584, 45882352941176466, 49411764705882348, 52941176470588230, 56470588235294112]
[4117647058823529, 8235294117647058, 12352941176470587, 16470588235294116, 20588235294117645, 24705882352941174, 28823529411764703, 32941176470588232, 37058823529411761, 41176470588235290, 45294117647058819, 49411764705882348, 53529411764705877, 57647058823529406, 61764705882352935, 65882352941176464]
[4705882352941176, 9411764705882352, 14117647058823528, 18823529411764704, 23529411764705880, 28235294117647056, 32941176470588232, 37647058823529408, 42352941176470584, 47058823529411760, 51764705882352936, 56470588235294112, 61176470588235288, 65882352941176464, 70588235294117640, 75294117647058816]
[5294117647058823, 10588235294117646, 15882352941176469, 21176470588235292, 26470588235294115, 31764705882352938, 37058823529411761, 42352941176470584, 47647058823529407, 52941176470588230, 58235294117647053, 63529411764705876, 68823529411764699, 74117647058823522, 79411764705882345, 84705882352941168]
[5882352941176470, 11764705882352940, 17647058823529410, 23529411764705880, 29411764705882350, 35294117647058820, 41176470588235290, 47058823529411760, 52941176470588230, 58823529411764700, 64705882352941170, 70588235294117640, 76470588235294110, 82352941176470580, 88235294117647050, 94117647058823520]
[6470588235294117, 12941176470588234, 19411764705882351, 25882352941176468, 32352941176470585, 38823529411764702, 45294117647058819, 51764705882352936, 58235294117647053, 64705882352941170, 71176470588235287, 77647058823529404, 84117647058823521, 90588235294117638, 97058823529411755, 103529411764705872]
[7058823529411764, 14117647058823528, 21176470588235292, 28235294117647056, 35294117647058820, 42352941176470584, 49411764705882348, 56470588235294112, 63529411764705876, 70588235294117640, 77647058823529404, 84705882352941168, 91764705882352932, 98823529411764696, 105882352941176460, 112941176470588224]
[7647058823529411, 15294117647058822, 22941176470588233, 30588235294117644, 38235294117647055, 45882352941176466, 53529411764705877, 61176470588235288, 68823529411764699, 76470588235294110, 84117647058823521, 91764705882352932, 99411764705882343, 107058823529411754, 114705882352941165, 122352941176470576]
[8235294117647058, 16470588235294116, 24705882352941174, 32941176470588232, 41176470588235290, 49411764705882348, 57647058823529406, 65882352941176464, 74117647058823522, 82352941176470580, 90588235294117638, 98823529411764696, 107058823529411754, 115294117647058812, 123529411764705870, 131764705882352928]
[8823529411764705, 17647058823529410, 26470588235294115, 35294117647058820, 44117647058823525, 52941176470588230, 61764705882352935, 70588235294117640, 79411764705882345, 88235294117647050, 97058823529411755, 105882352941176460, 114705882352941165, 123529411764705870, 132352941176470575, 141176470588235280]
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