181007

Ruby


Ramanujan's 54 universal quaternary quadratic forms(1)

n が100以下のときの表し方を出力してみた。

def s(k, n)
  s = 0
  (1..n).each{|i| s += i if n % i == 0 && i % k == 0}
  s
end

def A(ary, n)
  a_ary = [1]
  a = [0] + (1..n).map{|i| ary.inject(0){|s, j| s + j[1] * s(j[0], i)}}
  (1..n).each{|i| a_ary << (1..i).inject(0){|s, j| s - a[j] * a_ary[-j]} / i}
  a_ary
end

def B(a, n)
  ary = []
  a.each{|i|
    ary << [2 * i, 5]
    ary << [i, -2]
    ary << [4 * i, -2]
  }
  A(ary, n)
end

n = 100
a = []
(1.. 7).each{|i| a << [1, 1, 1, i]}
(2..14).each{|i| a << [1, 1, 2, i]}
(3.. 6).each{|i| a << [1, 1, 3, i]}
(2.. 7).each{|i| a << [1, 2, 2, i]}
(3..10).each{|i| a << [1, 2, 3, i]}
(4..14).each{|i| a << [1, 2, 4, i]}
(6..10).each{|i| a << [1, 2, 5, i]}
s = 0
a.each{|i|
  s += 1
  p [s, i]
  p B(i, n)
}

出力結果
[1, [1, 1, 1, 1]]
[1, 8, 24, 32, 24, 48, 96, 64, 24, 104, 144, 96, 96, 112, 192, 192, 24, 144, 312, 160, 144, 256, 288, 192, 96, 248, 336, 320, 192, 240, 576, 256, 24, 384, 432, 384, 312, 304, 480, 448, 144, 336, 768, 352, 288, 624, 576, 384, 96, 456, 744, 576, 336, 432, 960, 576, 192, 640, 720, 480, 576, 496, 768, 832, 24, 672, 1152, 544, 432, 768, 1152, 576, 312, 592, 912, 992, 480, 768, 1344, 640, 144, 968, 1008, 672, 768, 864, 1056, 960, 288, 720, 1872, 896, 576, 1024, 1152, 960, 96, 784, 1368, 1248, 744]
[2, [1, 1, 1, 2]]
[1, 6, 14, 20, 30, 40, 36, 48, 62, 42, 72, 100, 68, 120, 112, 48, 126, 108, 98, 180, 136, 160, 180, 144, 132, 126, 216, 200, 240, 280, 112, 192, 254, 120, 252, 320, 210, 360, 324, 144, 264, 252, 288, 420, 340, 280, 336, 288, 260, 342, 294, 360, 408, 520, 360, 240, 496, 216, 504, 580, 240, 600, 448, 336, 510, 288, 280, 660, 540, 480, 576, 432, 434, 444, 648, 420, 612, 800, 336, 480, 520, 366, 588, 820, 544, 720, 756, 336, 660, 540, 504, 960, 720, 640, 672, 432, 516, 588, 798, 700, 630]
[3, [1, 1, 1, 3]]
[1, 6, 12, 10, 18, 48, 40, 12, 60, 78, 24, 48, 70, 84, 120, 32, 66, 192, 84, 36, 144, 180, 120, 96, 136, 126, 168, 82, 84, 336, 200, 60, 252, 288, 96, 96, 234, 228, 360, 140, 120, 480, 144, 84, 336, 336, 240, 192, 310, 258, 252, 128, 252, 624, 400, 96, 408, 540, 168, 240, 224, 372, 600, 156, 258, 672, 360, 132, 576, 576, 240, 288, 420, 444, 456, 210, 252, 864, 560, 156, 528, 726, 240, 336, 540, 384, 840, 224, 408, 1056, 312, 168, 672, 900, 480, 288, 520, 588, 516, 336, 378]
[4, [1, 1, 1, 4]]
[1, 6, 12, 8, 8, 36, 48, 16, 24, 78, 72, 24, 32, 84, 96, 48, 24, 108, 156, 40, 48, 192, 144, 48, 96, 186, 168, 80, 64, 180, 288, 64, 24, 288, 216, 96, 104, 228, 240, 112, 144, 252, 384, 88, 96, 468, 288, 96, 96, 342, 372, 144, 112, 324, 480, 144, 192, 480, 360, 120, 192, 372, 384, 208, 24, 504, 576, 136, 144, 576, 576, 144, 312, 444, 456, 248, 160, 576, 672, 160, 144, 726, 504, 168, 256, 648, 528, 240, 288, 540, 936, 224, 192, 768, 576, 240, 96, 588, 684, 312, 248]
[5, [1, 1, 1, 5]]
[1, 6, 12, 8, 6, 26, 36, 24, 28, 42, 72, 72, 8, 48, 108, 48, 54, 64, 84, 120, 26, 72, 144, 88, 84, 126, 216, 80, 24, 180, 156, 192, 92, 96, 288, 144, 42, 144, 240, 144, 168, 252, 144, 168, 72, 182, 396, 184, 72, 258, 372, 192, 48, 208, 360, 312, 252, 160, 360, 360, 48, 372, 384, 168, 246, 288, 432, 264, 64, 264, 468, 432, 196, 288, 648, 248, 120, 288, 288, 480, 234, 366, 504, 328, 72, 384, 756, 240, 336, 540, 504, 432, 88, 256, 828, 520, 276, 384, 516, 504, 126]
[6, [1, 1, 1, 6]]
[1, 6, 12, 8, 6, 24, 26, 12, 36, 46, 36, 72, 56, 24, 72, 60, 54, 96, 52, 72, 120, 48, 36, 96, 98, 90, 144, 144, 60, 120, 144, 60, 180, 172, 96, 240, 142, 72, 192, 96, 132, 288, 180, 120, 216, 168, 144, 192, 152, 162, 276, 256, 120, 216, 210, 120, 264, 256, 180, 360, 300, 120, 360, 156, 150, 480, 208, 264, 384, 240, 168, 288, 316, 252, 432, 440, 216, 240, 360, 156, 504, 374, 144, 552, 240, 192, 384, 300, 228, 576, 468, 240, 480, 240, 288, 480, 434, 300, 420, 440, 378]
[7, [1, 1, 1, 7]]
[1, 6, 12, 8, 6, 24, 24, 2, 24, 54, 40, 36, 56, 72, 48, 24, 66, 96, 84, 40, 72, 144, 24, 12, 120, 102, 120, 80, 98, 132, 72, 64, 84, 240, 160, 48, 198, 180, 120, 72, 136, 240, 240, 84, 84, 312, 120, 96, 248, 246, 180, 96, 216, 228, 240, 80, 156, 360, 216, 120, 168, 360, 192, 194, 174, 216, 400, 36, 288, 528, 144, 84, 336, 432, 312, 168, 280, 396, 216, 180, 264, 558, 400, 168, 432, 288, 168, 240, 288, 528, 520, 144, 252, 576, 288, 120, 504, 576, 396, 132, 246]
[8, [1, 1, 2, 2]]
[1, 4, 8, 16, 24, 24, 32, 32, 24, 52, 48, 48, 96, 56, 64, 96, 24, 72, 104, 80, 144, 128, 96, 96, 96, 124, 112, 160, 192, 120, 192, 128, 24, 192, 144, 192, 312, 152, 160, 224, 144, 168, 256, 176, 288, 312, 192, 192, 96, 228, 248, 288, 336, 216, 320, 288, 192, 320, 240, 240, 576, 248, 256, 416, 24, 336, 384, 272, 432, 384, 384, 288, 312, 296, 304, 496, 480, 384, 448, 320, 144, 484, 336, 336, 768, 432, 352, 480, 288, 360, 624, 448, 576, 512, 384, 480, 96, 392, 456, 624, 744]
[9, [1, 1, 2, 3]]
[1, 4, 6, 10, 20, 20, 24, 40, 22, 28, 56, 20, 50, 80, 28, 80, 84, 32, 78, 80, 68, 100, 120, 80, 88, 124, 56, 82, 136, 100, 140, 200, 86, 80, 192, 72, 140, 240, 120, 200, 248, 80, 112, 176, 100, 260, 224, 160, 210, 172, 186, 128, 272, 180, 240, 400, 124, 200, 280, 116, 272, 400, 140, 280, 340, 144, 300, 272, 160, 320, 432, 240, 286, 296, 168, 310, 400, 200, 224, 520, 260, 244, 480, 164, 340, 640, 264, 400, 440, 176, 392, 288, 272, 500, 448, 400, 344, 392, 258, 260, 620]
[10, [1, 1, 2, 4]]
[1, 4, 6, 8, 14, 16, 20, 32, 30, 28, 40, 40, 36, 48, 48, 32, 62, 72, 42, 72, 72, 64, 100, 96, 68, 84, 120, 80, 112, 112, 48, 128, 126, 80, 108, 128, 98, 144, 180, 96, 136, 168, 160, 168, 180, 112, 144, 192, 132, 228, 126, 144, 216, 208, 200, 160, 240, 144, 280, 232, 112, 240, 192, 224, 254, 192, 120, 264, 252, 192, 320, 288, 210, 296, 360, 168, 324, 320, 144, 320, 264, 244, 252, 328, 288, 288, 420, 224, 340, 360, 280, 384, 336, 256, 288, 288, 260, 392, 342, 280, 294]
[11, [1, 1, 2, 5]]
[1, 4, 6, 8, 12, 10, 16, 28, 22, 36, 40, 24, 56, 36, 24, 64, 28, 64, 78, 24, 58, 48, 68, 92, 80, 92, 72, 112, 92, 48, 104, 96, 118, 176, 64, 48, 124, 84, 148, 160, 104, 120, 176, 120, 72, 146, 88, 204, 216, 124, 126, 96, 180, 148, 224, 188, 120, 304, 216, 120, 224, 96, 192, 332, 156, 240, 160, 168, 224, 144, 196, 192, 350, 304, 168, 232, 168, 168, 336, 224, 234, 340, 252, 200, 192, 192, 240, 480, 292, 264, 392, 144, 316, 320, 184, 332, 304, 416, 258, 216, 308]
[12, [1, 1, 2, 6]]
[1, 4, 6, 8, 12, 8, 10, 24, 18, 28, 48, 24, 40, 56, 12, 40, 60, 32, 78, 72, 24, 48, 48, 48, 70, 84, 84, 80, 120, 56, 32, 120, 66, 120, 192, 48, 84, 152, 36, 112, 144, 80, 180, 168, 120, 104, 96, 96, 136, 172, 126, 160, 168, 104, 82, 192, 84, 144, 336, 120, 200, 248, 60, 168, 252, 112, 288, 264, 96, 240, 96, 144, 234, 296, 228, 168, 360, 144, 140, 312, 120, 244, 480, 168, 144, 256, 84, 280, 336, 176, 336, 336, 240, 240, 192, 144, 310, 392, 258, 312, 252]
[13, [1, 1, 2, 7]]
[1, 4, 6, 8, 12, 8, 8, 18, 14, 24, 40, 32, 40, 40, 32, 28, 36, 64, 46, 72, 72, 16, 56, 40, 40, 88, 72, 80, 82, 64, 44, 64, 86, 96, 128, 144, 64, 64, 136, 44, 136, 160, 80, 144, 128, 120, 92, 128, 104, 164, 158, 96, 168, 80, 144, 160, 168, 108, 104, 232, 108, 168, 192, 82, 156, 252, 160, 208, 192, 80, 272, 132, 134, 288, 152, 248, 232, 200, 124, 148, 328, 192, 288, 328, 144, 160, 264, 160, 248, 352, 216, 336, 184, 128, 256, 188, 232, 384, 278, 240, 320]
[14, [1, 1, 2, 8]]
[1, 4, 6, 8, 12, 8, 8, 16, 8, 20, 36, 24, 48, 40, 16, 48, 24, 40, 78, 40, 72, 64, 24, 48, 32, 60, 84, 80, 96, 40, 48, 64, 24, 96, 108, 96, 156, 72, 40, 112, 48, 104, 192, 88, 144, 168, 48, 96, 96, 100, 186, 144, 168, 136, 80, 144, 64, 160, 180, 120, 288, 104, 64, 208, 24, 144, 288, 136, 216, 192, 96, 144, 104, 136, 228, 248, 240, 192, 112, 160, 144, 260, 252, 168, 384, 208, 88, 240, 96, 200, 468, 224, 288, 256, 96, 240, 96, 232, 342, 312, 372]
[15, [1, 1, 2, 9]]
[1, 4, 6, 8, 12, 8, 8, 16, 6, 14, 32, 20, 40, 48, 16, 32, 44, 28, 54, 72, 40, 64, 72, 16, 40, 52, 56, 92, 96, 56, 32, 80, 38, 48, 104, 64, 102, 144, 52, 32, 112, 44, 128, 168, 68, 136, 128, 96, 104, 132, 62, 144, 176, 104, 172, 96, 112, 112, 192, 116, 96, 240, 96, 176, 204, 32, 112, 264, 108, 192, 224, 48, 198, 168, 104, 168, 232, 160, 160, 208, 104, 178, 232, 164, 192, 288, 164, 96, 264, 140, 232, 384, 144, 256, 288, 112, 232, 232, 150, 340, 252]
[16, [1, 1, 2, 10]]
[1, 4, 6, 8, 12, 8, 8, 16, 6, 12, 26, 16, 36, 40, 24, 32, 28, 48, 42, 48, 72, 32, 72, 64, 8, 60, 48, 64, 108, 56, 48, 64, 54, 64, 64, 72, 84, 104, 120, 32, 26, 88, 72, 120, 144, 88, 88, 144, 84, 92, 126, 64, 216, 152, 80, 160, 24, 128, 180, 144, 156, 104, 192, 128, 92, 168, 96, 184, 288, 96, 144, 160, 42, 224, 144, 104, 240, 224, 144, 128, 168, 100, 252, 248, 144, 208, 168, 176, 72, 168, 182, 160, 396, 192, 184, 224, 72, 272, 258, 176, 372]
[17, [1, 1, 2, 11]]
[1, 4, 6, 8, 12, 8, 8, 16, 6, 12, 24, 10, 32, 36, 16, 40, 28, 32, 62, 36, 48, 64, 40, 64, 56, 28, 56, 56, 80, 68, 48, 80, 38, 80, 80, 32, 92, 88, 88, 112, 40, 16, 112, 52, 90, 144, 44, 120, 128, 100, 78, 96, 116, 144, 160, 112, 128, 80, 136, 104, 144, 132, 92, 240, 108, 112, 192, 88, 128, 232, 144, 176, 206, 80, 120, 128, 116, 192, 208, 160, 160, 212, 176, 132, 224, 128, 200, 272, 136, 112, 136, 160, 168, 312, 140, 144, 248, 176, 214, 186, 156]
[18, [1, 1, 2, 12]]
[1, 4, 6, 8, 12, 8, 8, 16, 6, 12, 24, 8, 26, 32, 12, 32, 36, 32, 46, 56, 36, 40, 72, 32, 56, 76, 24, 64, 72, 40, 60, 80, 54, 64, 96, 48, 52, 96, 72, 80, 120, 32, 48, 104, 36, 104, 96, 64, 98, 124, 90, 64, 144, 72, 144, 160, 60, 136, 120, 104, 144, 160, 60, 112, 180, 96, 172, 200, 96, 128, 240, 96, 142, 200, 72, 200, 192, 80, 96, 208, 132, 164, 288, 104, 180, 256, 120, 160, 216, 128, 168, 192, 144, 200, 192, 160, 152, 248, 162, 168, 276]
[19, [1, 1, 2, 13]]
[1, 4, 6, 8, 12, 8, 8, 16, 6, 12, 24, 8, 24, 26, 8, 28, 28, 40, 46, 40, 56, 28, 48, 64, 24, 76, 72, 32, 80, 32, 32, 92, 54, 80, 80, 64, 68, 56, 80, 64, 88, 112, 64, 40, 88, 52, 64, 128, 56, 108, 110, 64, 106, 80, 76, 160, 104, 96, 168, 56, 124, 112, 120, 124, 92, 208, 96, 152, 168, 64, 172, 208, 126, 192, 152, 72, 184, 112, 80, 192, 232, 148, 160, 200, 92, 172, 248, 128, 176, 256, 184, 80, 160, 128, 168, 288, 136, 256, 150, 152, 228]
[20, [1, 1, 2, 14]]
[1, 4, 6, 8, 12, 8, 8, 16, 6, 12, 24, 8, 24, 24, 2, 24, 24, 32, 54, 40, 40, 48, 36, 40, 56, 44, 72, 80, 48, 40, 24, 64, 66, 80, 96, 48, 84, 72, 40, 72, 72, 80, 144, 56, 24, 104, 12, 96, 120, 68, 102, 96, 120, 56, 80, 80, 98, 120, 132, 120, 72, 120, 64, 152, 84, 72, 240, 120, 160, 176, 48, 112, 198, 144, 180, 168, 120, 160, 72, 96, 136, 204, 240, 168, 240, 96, 84, 240, 84, 176, 312, 144, 120, 192, 96, 120, 248, 192, 246, 216, 180]
[21, [1, 1, 3, 3]]
[1, 4, 4, 4, 20, 24, 4, 32, 52, 4, 24, 48, 20, 56, 32, 24, 116, 72, 4, 80, 120, 32, 48, 96, 52, 124, 56, 4, 160, 120, 24, 128, 244, 48, 72, 192, 20, 152, 80, 56, 312, 168, 32, 176, 240, 24, 96, 192, 116, 228, 124, 72, 280, 216, 4, 288, 416, 80, 120, 240, 120, 248, 128, 32, 500, 336, 48, 272, 360, 96, 192, 288, 52, 296, 152, 124, 400, 384, 56, 320, 696, 4, 168, 336, 160, 432, 176, 120, 624, 360, 24, 448, 480, 128, 192, 480, 244, 392, 228, 48, 620]
[22, [1, 1, 3, 4]]
[1, 4, 4, 2, 14, 24, 8, 12, 44, 36, 8, 24, 50, 40, 24, 16, 50, 96, 28, 12, 112, 96, 24, 48, 104, 68, 56, 50, 60, 168, 40, 12, 188, 144, 32, 48, 182, 104, 72, 76, 88, 240, 48, 60, 240, 168, 48, 96, 230, 124, 84, 64, 196, 312, 80, 48, 312, 264, 56, 120, 160, 200, 120, 60, 194, 336, 120, 60, 448, 288, 48, 144, 308, 232, 152, 90, 180, 432, 112, 108, 400, 372, 80, 168, 420, 192, 168, 112, 312, 528, 104, 72, 480, 432, 96, 144, 392, 280, 172, 168, 294]
[23, [1, 1, 3, 5]]
[1, 4, 4, 2, 12, 18, 8, 16, 24, 28, 40, 8, 26, 72, 16, 16, 44, 44, 68, 24, 34, 80, 72, 28, 40, 124, 40, 50, 112, 56, 80, 40, 76, 144, 120, 32, 84, 216, 24, 40, 136, 80, 160, 88, 56, 154, 88, 28, 158, 228, 100, 48, 216, 172, 80, 104, 80, 300, 280, 40, 112, 248, 120, 112, 172, 144, 200, 136, 172, 176, 208, 48, 200, 432, 120, 90, 168, 288, 144, 104, 118, 244, 400, 76, 240, 432, 88, 112, 360, 176, 280, 128, 172, 420, 184, 108, 168, 576, 292, 104, 372]
[24, [1, 1, 3, 6]]
[1, 4, 4, 2, 12, 16, 2, 16, 28, 8, 32, 40, 10, 48, 48, 8, 44, 40, 24, 72, 48, 16, 72, 64, 10, 52, 80, 38, 96, 112, 8, 80, 108, 12, 104, 128, 48, 144, 136, 8, 112, 104, 32, 168, 136, 64, 128, 96, 26, 132, 116, 36, 176, 208, 70, 96, 192, 28, 192, 232, 24, 240, 176, 80, 204, 128, 28, 264, 216, 48, 224, 192, 96, 168, 272, 42, 232, 320, 40, 208, 208, 76, 232, 328, 48, 288, 296, 24, 264, 200, 112, 384, 288, 64, 288, 160, 58, 232, 324, 160, 252]
[25, [1, 2, 2, 2]]
[1, 2, 6, 12, 14, 24, 20, 16, 30, 14, 40, 60, 36, 72, 48, 16, 62, 36, 42, 108, 72, 96, 100, 48, 68, 42, 120, 120, 112, 168, 48, 64, 126, 40, 108, 192, 98, 216, 180, 48, 136, 84, 160, 252, 180, 168, 144, 96, 132, 114, 126, 216, 216, 312, 200, 80, 240, 72, 280, 348, 112, 360, 192, 112, 254, 96, 120, 396, 252, 288, 320, 144, 210, 148, 360, 252, 324, 480, 144, 160, 264, 122, 252, 492, 288, 432, 420, 112, 340, 180, 280, 576, 336, 384, 288, 144, 260, 196, 342, 420, 294]
[26, [1, 2, 2, 3]]
[1, 2, 4, 10, 10, 16, 24, 12, 28, 26, 8, 48, 30, 28, 72, 32, 34, 64, 28, 36, 80, 60, 72, 96, 72, 42, 56, 82, 36, 112, 120, 60, 124, 96, 32, 96, 130, 76, 216, 140, 56, 160, 48, 84, 144, 112, 144, 192, 150, 86, 84, 128, 140, 208, 240, 96, 216, 180, 56, 240, 96, 124, 360, 156, 130, 224, 120, 132, 320, 192, 144, 288, 196, 148, 152, 210, 108, 288, 336, 156, 272, 242, 80, 336, 300, 128, 504, 224, 216, 352, 104, 168, 288, 300, 288, 288, 264, 196, 172, 336, 210]
[27, [1, 2, 2, 4]]
[1, 2, 4, 8, 8, 12, 16, 16, 24, 26, 24, 24, 32, 28, 32, 48, 24, 36, 52, 40, 48, 64, 48, 48, 96, 62, 56, 80, 64, 60, 96, 64, 24, 96, 72, 96, 104, 76, 80, 112, 144, 84, 128, 88, 96, 156, 96, 96, 96, 114, 124, 144, 112, 108, 160, 144, 192, 160, 120, 120, 192, 124, 128, 208, 24, 168, 192, 136, 144, 192, 192, 144, 312, 148, 152, 248, 160, 192, 224, 160, 144, 242, 168, 168, 256, 216, 176, 240, 288, 180, 312, 224, 192, 256, 192, 240, 96, 196, 228, 312, 248]
[28, [1, 2, 2, 5]]
[1, 2, 4, 8, 6, 10, 12, 8, 28, 22, 24, 40, 8, 32, 36, 16, 54, 32, 28, 56, 26, 32, 48, 40, 84, 74, 72, 48, 24, 92, 52, 96, 92, 32, 96, 80, 42, 64, 80, 80, 168, 124, 48, 72, 72, 94, 132, 104, 72, 126, 124, 96, 48, 96, 120, 168, 252, 96, 120, 168, 48, 196, 128, 88, 246, 128, 144, 104, 64, 128, 156, 208, 196, 160, 216, 120, 120, 160, 96, 256, 234, 194, 168, 168, 72, 192, 252, 112, 336, 276, 168, 208, 88, 128, 276, 248, 276, 160, 172, 248, 126]
[29, [1, 2, 2, 6]]
[1, 2, 4, 8, 6, 8, 10, 4, 20, 26, 20, 40, 24, 8, 40, 20, 22, 64, 28, 40, 56, 16, 20, 32, 50, 62, 80, 80, 28, 40, 80, 20, 84, 100, 32, 144, 78, 24, 80, 32, 68, 160, 100, 88, 120, 56, 80, 64, 88, 86, 124, 160, 56, 72, 82, 40, 136, 160, 100, 232, 140, 40, 200, 52, 86, 288, 80, 136, 192, 80, 72, 96, 140, 148, 240, 248, 120, 80, 200, 52, 248, 242, 80, 328, 112, 64, 176, 100, 100, 352, 260, 144, 224, 80, 160, 160, 210, 196, 172, 280, 186]
[30, [1, 2, 2, 7]]
[1, 2, 4, 8, 6, 8, 8, 2, 16, 18, 24, 36, 24, 24, 16, 24, 26, 32, 60, 40, 40, 48, 8, 12, 56, 34, 72, 80, 34, 44, 24, 64, 44, 80, 96, 48, 102, 60, 40, 72, 72, 80, 144, 84, 52, 104, 40, 96, 120, 82, 92, 96, 120, 76, 80, 80, 84, 120, 136, 120, 72, 120, 64, 194, 102, 72, 240, 36, 160, 176, 48, 84, 168, 144, 168, 168, 120, 132, 72, 180, 136, 186, 240, 168, 240, 96, 56, 240, 112, 176, 312, 144, 92, 192, 96, 120, 248, 192, 260, 132, 150]
[31, [1, 2, 3, 3]]
[1, 2, 2, 8, 10, 8, 24, 16, 10, 38, 8, 12, 48, 8, 32, 64, 26, 36, 70, 28, 24, 80, 28, 48, 96, 42, 40, 76, 48, 24, 112, 64, 58, 160, 68, 32, 126, 56, 44, 192, 56, 84, 176, 44, 60, 88, 80, 96, 208, 114, 74, 176, 72, 72, 172, 80, 112, 288, 88, 76, 224, 72, 112, 304, 90, 96, 288, 60, 116, 224, 96, 144, 254, 148, 88, 200, 108, 96, 352, 160, 88, 362, 148, 124, 384, 80, 124, 448, 124, 180, 216, 128, 176, 304, 192, 144, 400, 196, 210, 228, 146]
[32, [1, 2, 3, 4]]
[1, 2, 2, 6, 8, 8, 16, 16, 14, 22, 24, 16, 22, 32, 12, 32, 44, 16, 42, 52, 36, 40, 64, 32, 40, 86, 24, 50, 72, 40, 60, 80, 38, 48, 112, 48, 72, 96, 64, 80, 120, 64, 48, 124, 52, 104, 96, 64, 106, 110, 110, 96, 144, 72, 128, 160, 60, 132, 120, 64, 144, 160, 60, 112, 164, 96, 164, 172, 64, 128, 240, 96, 150, 196, 72, 210, 184, 80, 96, 208, 132, 162, 256, 112, 180, 256, 160, 160, 232, 112, 168, 192, 144, 200, 192, 160, 184, 268, 134, 176, 296]
[33, [1, 2, 3, 5]]
[1, 2, 2, 6, 6, 6, 16, 8, 14, 26, 8, 32, 26, 8, 40, 16, 22, 40, 22, 32, 46, 40, 24, 48, 40, 42, 72, 50, 32, 64, 56, 28, 74, 48, 60, 112, 78, 24, 72, 76, 40, 144, 48, 48, 120, 50, 52, 48, 70, 98, 150, 128, 40, 84, 128, 52, 176, 120, 56, 208, 96, 72, 92, 72, 102, 192, 156, 80, 120, 96, 104, 160, 106, 80, 184, 178, 120, 112, 96, 76, 210, 210, 80, 196, 208, 80, 256, 80, 72, 304, 136, 208, 136, 132, 116, 160, 184, 112, 258, 256, 126]
[34, [1, 2, 3, 6]]
[1, 2, 2, 6, 6, 4, 14, 8, 6, 26, 12, 16, 42, 12, 16, 44, 6, 20, 50, 16, 36, 56, 24, 16, 42, 30, 28, 78, 48, 36, 84, 40, 6, 80, 36, 48, 150, 44, 40, 100, 36, 36, 112, 48, 72, 148, 48, 48, 42, 50, 62, 124, 84, 52, 158, 64, 48, 144, 60, 64, 252, 60, 64, 200, 6, 88, 168, 64, 108, 176, 96, 80, 150, 84, 76, 218, 120, 96, 196, 72, 36, 242, 84, 80, 336, 104, 88, 204, 72, 84, 300, 112, 144, 216, 96, 128, 42, 100, 114, 304, 186]
[35, [1, 2, 3, 7]]
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[36, [1, 2, 3, 8]]
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[37, [1, 2, 3, 9]]
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[38, [1, 2, 3, 10]]
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[39, [1, 2, 4, 4]]
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[40, [1, 2, 4, 5]]
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[41, [1, 2, 4, 6]]
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[42, [1, 2, 4, 7]]
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[43, [1, 2, 4, 8]]
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[44, [1, 2, 4, 9]]
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[45, [1, 2, 4, 10]]
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[46, [1, 2, 4, 11]]
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[47, [1, 2, 4, 12]]
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[48, [1, 2, 4, 13]]
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[49, [1, 2, 4, 14]]
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[50, [1, 2, 5, 6]]
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[51, [1, 2, 5, 7]]
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[52, [1, 2, 5, 8]]
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[53, [1, 2, 5, 9]]
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[54, [1, 2, 5, 10]]
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