240330

PARI


Continued fraction for Pi^n.

Pi^4の連分数展開が興味深い。

default(realprecision, 70);
for(n=1, 100, print([n, contfrac(Pi^n)]))

出力結果

[1, [3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, 84, 2, 1, 1, 15, 3, 13, 1, 4, 2, 6, 6, 99, 1, 2, 2, 6, 3, 5, 1, 1, 6, 8, 1, 7, 1, 2, 3, 7, 1, 2, 1, 1, 12, 1, 1, 1, 3, 1, 1, 8, 1, 1, 2, 1, 6, 1, 1, 5, 2, 2, 3, 1, 2, 4, 4]]

[2, [9, 1, 6, 1, 2, 47, 1, 8, 1, 1, 2, 2, 1, 1, 8, 3, 1, 10, 5, 1, 3, 1, 2, 1, 1, 3, 15, 1, 1, 2, 2, 1, 3, 2, 7, 1, 9, 18, 30, 2, 145, 1, 1, 17, 9, 1, 1, 1, 1, 7, 12, 1, 2, 1, 12, 1, 1, 4, 1, 5, 1, 1, 2, 3, 4, 1, 3, 2, 9, 1, 20, 11, 14]]

[3, [31, 159, 3, 7, 1, 13, 2, 1, 3, 1, 12, 2, 2, 4, 34, 2, 43, 3, 1, 3, 2, 1, 1, 5, 1, 1, 4, 1, 5, 4, 2, 4, 11, 3, 3, 1, 1, 2, 1, 7, 2, 1, 1, 3, 1, 12, 3, 1, 9, 2, 1, 8, 23, 1, 45, 1, 1, 2, 1, 23, 3, 2, 2, 2, 1, 2, 2, 1, 1, 2, 2, 1, 16]]

[4, [97, 2, 2, 3, 1, 16539, 1, 6, 7, 6, 8, 6, 3, 9, 1, 1, 1, 18, 1, 4, 1, 13, 1, 2, 1, 127, 1, 1, 1, 4, 1, 6, 1, 1, 1, 10, 10, 1, 1, 2, 1, 2, 1, 5, 1, 1, 10, 1, 3, 2, 1, 1, 4, 9, 1, 7, 70, 1, 13, 1, 2, 6, 1, 2, 24, 5, 2]]

[5, [306, 50, 1, 4, 60, 1, 4, 23, 1, 2, 1, 1, 1, 17, 1, 5, 1, 1, 1, 2, 4, 10, 4, 1, 3, 2, 1, 36, 7, 1, 27, 3, 1, 6, 1, 2, 1, 1, 3, 1, 1, 1, 53, 1, 1, 1, 15, 1, 3, 573, 1, 5, 2, 1, 4, 1, 3, 2, 1, 15, 2, 1, 1, 1, 4, 3, 1, 66]]

[6, [961, 2, 1, 1, 3, 9, 1, 5, 1, 3, 6, 1, 3, 1, 77, 11, 3, 1, 11, 14, 4, 4, 27, 5, 178, 5, 1, 2, 2, 1099, 2, 3, 71, 4, 1, 3, 1, 1, 1, 2, 16, 154, 1, 3, 1, 5, 4, 5, 1, 1, 17, 1, 1, 1, 1, 1, 2, 4]]

[7, [3020, 3, 2, 2, 3, 2, 10, 4, 1, 1, 2, 1, 3, 13, 58, 2, 50, 1, 2, 21, 1, 1, 2, 1, 14, 1, 182, 2, 1, 1, 2, 69, 6, 2, 4, 3, 2, 5, 11, 2, 2, 5, 4, 1, 5, 41, 1, 2, 3, 1, 2, 1, 2, 1, 1, 1, 1, 1, 8, 1, 5, 1, 2, 2, 3]]

[8, [9488, 1, 1, 7, 1, 1, 3, 1, 1, 1, 4, 10, 13, 1, 1, 2, 3, 6, 1, 1, 1, 135, 1, 2, 2, 3, 1, 2, 1, 7, 3, 1, 1, 4, 2, 7, 1, 1, 1, 27, 4, 1, 3, 13, 1, 3, 2, 17, 1, 1, 3, 3, 2, 1, 1, 1, 1, 1, 6, 5, 5, 5, 1, 2, 1, 1, 1, 2, 2, 2, 1, 6, 11, 2, 24, 3, 25, 3]]

[9, [29809, 10, 14, 1, 9, 3, 1, 5, 8, 16, 3, 15, 7, 39, 3, 3, 1, 1, 2, 1, 3, 4, 1, 1, 7, 2, 1, 2, 1, 3, 10, 5, 72, 1, 1, 3, 2, 1, 2, 2, 6, 3, 7, 2, 1, 3, 6, 1, 6, 2, 3, 20, 1, 2, 8, 9, 1, 14, 3, 1, 2, 4, 14]]

[10, [93648, 21, 15, 1, 4, 2, 1, 3, 2, 1, 4, 2, 74, 26, 5, 3, 1, 5, 17, 1, 2, 1, 1, 2, 1, 14, 2, 2, 10, 7, 4, 2, 2, 3, 1, 2, 1, 3, 3, 1, 1, 1, 23, 13, 2, 2, 1, 1, 2, 13, 9, 9, 1, 6, 1, 2, 1, 7, 2, 2, 12, 6, 1, 16, 8]]

[11, [294204, 55, 1, 1, 1, 2, 1, 76, 1, 10, 1, 1, 1, 1, 10, 1, 41, 3, 1, 1, 2, 2, 3, 6, 6, 2, 1, 5, 1, 1, 1, 12, 1, 61, 6, 2, 44, 1, 8, 1, 5, 5, 1, 2, 1, 5, 3, 1, 2, 8, 4, 1, 2, 3, 2, 2, 1, 1, 3, 1, 21, 2, 1, 2, 1, 1, 1, 3, 5, 1, 1, 1, 1, 6]]

[12, [924269, 5, 1, 1, 27, 2, 20, 3, 1, 15, 3, 1, 1, 2, 2, 1, 5, 4, 3, 2, 1, 1, 6, 64, 9, 5, 3, 1, 8, 1, 1, 2, 1, 2, 8, 1, 2, 1, 1, 1, 2, 4, 1, 1, 26, 1, 50, 2, 1, 3, 1, 1, 19, 1, 4, 3, 6, 1, 1, 1, 15, 1, 3, 1, 1, 2, 1, 1, 7, 5, 1, 1, 1, 1, 5]]

[13, [2903677, 3, 1, 2, 3, 1, 1, 4, 1, 24, 1, 1, 1, 2, 1, 26, 1, 4, 2, 1, 5, 1, 1, 6, 1, 7, 22, 2, 1, 7, 5, 1, 4, 1, 1, 1, 1, 1, 6, 1, 1, 1, 5, 1, 2, 1, 2, 146, 2, 1, 1, 2, 2, 3, 5, 3, 2, 1, 1, 1, 24, 2, 2, 3, 2, 14, 1, 2, 1, 5, 11, 4, 3, 2, 1, 1, 5]]

[14, [9122171, 5, 1, 1, 128, 1, 14, 41, 2, 2, 1, 1, 1, 3, 1, 9, 4, 1, 1, 1, 144, 2, 1, 14, 3, 2, 10, 1, 1, 2044, 1, 2, 2, 2, 1, 3, 5, 1, 1, 1, 1, 1, 5, 404, 1, 1, 1, 1, 16, 1, 2, 21, 1, 1, 3, 46, 10, 12]]

[15, [28658145, 1, 31, 1, 2, 427, 1, 1, 6, 1, 1, 5, 1, 1, 4, 3, 14, 5, 7, 2, 1, 4, 1, 8, 6, 1, 2, 1, 1, 2, 1, 6, 15, 8, 21, 1, 1, 1, 11, 2, 2, 3, 1, 3, 2, 1, 3, 101, 1, 2, 3, 5, 1, 1, 10, 45, 4, 1, 4, 1, 3, 10, 3]]

[16, [90032220, 1, 5, 2, 1, 2, 1, 1, 1, 19, 1, 3, 1, 2, 7, 2, 2, 1, 1, 2, 8, 2, 1, 4, 17, 1, 7, 1, 1, 1, 1, 1, 1, 4, 1, 1, 2, 2, 1, 12, 18, 1, 2, 4, 2, 5, 6, 1, 14, 1, 1, 1, 13, 1, 9, 1, 1, 33, 1, 1, 1, 2, 2, 2, 21, 3, 5, 3, 1, 1, 1, 4, 1, 2, 1, 1, 2, 4, 4]]

[17, [282844563, 1, 1, 2, 2, 1, 1, 3, 16, 1, 4, 2, 3, 2, 1, 2, 13, 2, 2, 6, 12, 5, 1, 3, 1, 1, 6, 1, 1, 1, 1, 3, 4, 3, 2, 2, 1, 3, 4, 2, 3, 1, 6, 15, 1, 2, 1, 1, 1, 13, 1, 1, 1, 7, 2, 20, 2, 1, 2, 2, 9, 3, 1, 2, 6, 1, 1, 3, 5, 1, 21, 1, 4, 3, 1, 4]]

[18, [888582403, 14, 30, 1, 4, 2, 1, 1, 2, 5, 152, 3, 1, 6, 5, 7, 2, 1, 9, 1, 1, 1, 267, 1, 1, 3, 37, 1, 5, 1, 1, 4, 1, 2, 2, 9, 1, 4, 4, 70, 1, 2, 1, 3, 1, 66, 4, 1, 2, 3, 1, 8, 2, 7, 1, 2, 9, 3, 6]]

[19, [2791563949, 1, 1, 2, 18, 6, 89, 163, 1, 68, 5, 1, 1, 2, 1, 4, 29, 13, 1, 1, 1, 1, 1, 3, 1, 2, 2, 70, 4, 1, 3, 6, 1, 2, 1, 4, 1, 3, 5, 5, 1, 2, 1, 1, 3, 3, 1, 3, 1, 6, 1, 1, 1, 1, 3, 5, 2, 1, 3, 2, 2, 1, 1, 1, 2, 6, 6]]

[20, [8769956796, 12, 10, 1, 6, 1, 5, 2, 1, 3, 1, 93, 1, 1, 2, 1, 12, 4, 3, 4, 5, 2, 2, 3, 6, 1, 1, 2, 14, 1, 14, 1, 1, 2, 2, 12, 44, 1, 1, 1, 6, 1, 1, 1, 4, 1, 3, 3, 4, 4, 2, 1, 4, 1, 9, 1, 3, 7, 6, 1, 1, 1, 2, 1, 2, 2, 1, 2, 3, 2]]

[21, [27551631842, 1, 6, 1, 8, 3, 1, 7, 1, 1, 1, 2, 9, 1, 23, 1, 2, 649, 2, 2, 1, 8, 1, 2, 6, 1, 1, 1, 153, 2, 2, 743, 1, 22, 1, 2, 1, 11, 1, 3, 11, 2, 1, 1, 3, 1, 1, 1, 6, 1, 3, 4, 5, 1, 3, 2, 1, 2, 1, 16, 12]]

[22, [86556004191, 1, 52, 1, 1, 2, 7, 1, 1, 36, 1, 7, 2, 1, 57, 18, 11, 401, 2, 46, 2, 2, 1, 1, 1, 1, 3, 1, 2, 2, 1, 2, 19, 23, 2, 2, 1, 9, 4, 1, 1, 2, 2, 2, 7, 3, 3, 1, 2, 1, 34, 5, 1, 1, 4, 25]]

[23, [271923706893, 1, 1, 1, 1, 1, 1, 10, 4, 3, 2, 2, 3, 1, 26, 163, 1, 80, 1, 1, 1, 1, 3, 2, 1, 2, 1, 2, 12, 2, 1, 1, 5, 1, 2, 10, 3, 2, 1, 5, 1, 16, 1, 51, 4, 1, 2, 2, 1, 8, 1, 1, 6, 2, 2, 1, 1, 1, 1, 1, 2, 3, 14, 1, 2, 1, 1, 29]]

[24, [854273519913, 1, 7, 1, 13, 2, 1, 4, 2, 1, 44, 1, 1, 4, 2, 1, 2, 4, 1, 1, 1, 6, 1, 4, 1, 1, 3, 1, 1, 1, 5, 3, 3, 31, 1, 1, 1, 5, 2, 1, 2, 1, 26, 8, 1, 2, 2, 8, 1, 7, 2, 1, 1, 1, 1, 1, 1, 1, 6, 2, 1, 23, 5, 2, 3, 1, 14, 1, 9]]

[25, [2683779414317, 1, 3, 4, 21, 1, 4, 1, 1, 1, 1, 1, 8, 1, 1, 6, 1, 49, 1, 1, 13, 3, 1, 4, 1, 3, 7, 2, 8, 1, 5, 2, 1, 2, 3, 7, 1, 1, 2, 1, 1, 8, 1, 2, 1, 11, 1, 12, 1, 2, 1, 4, 9, 3, 1, 2, 19, 1, 1, 7, 1, 2, 2, 1, 1, 1, 4, 3, 2, 1, 5, 1, 1, 3]]

[26, [8431341691876, 4, 1, 4, 1, 6, 6, 7, 1, 2, 26, 1, 3, 1, 7, 4, 4, 1, 1, 1, 1, 1, 3, 1, 2, 4, 1, 1, 14, 65, 2, 3, 12, 1, 1, 3, 3, 2, 11, 5, 11, 1, 1, 4, 2, 2, 1, 9, 1, 1, 1, 1, 1, 18, 1, 1, 1, 2, 2, 3, 1, 2, 2, 1, 5, 1, 2, 2, 1, 1, 2]]

[27, [26487841119103, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 4, 1, 2, 2, 6, 2, 97, 1, 1, 7, 1, 1, 1, 1, 1, 3, 5, 1, 2, 1, 9, 1, 2, 1, 1, 5, 3, 57, 1, 4, 2, 11, 1, 1, 16, 6, 2, 2, 3, 1, 1, 1, 5, 20, 9, 4, 6, 9, 1, 1, 1, 3, 13, 1, 3, 7]]

[28, [83214007069229, 1, 1, 1, 1, 2, 1, 1, 1, 25, 1, 4, 1, 4, 1, 34, 17, 2, 1, 3, 1, 4, 1, 3, 5, 1, 5, 6, 4, 11, 3, 23, 1, 2, 3, 2, 5, 1, 1, 1, 6, 3, 3, 1, 1, 13, 1, 1, 10, 1, 16, 1, 1, 906, 1, 2, 27, 36]]

[29, [261424513284460, 1, 6, 1, 2, 1, 212, 1, 19, 1, 3, 1, 169, 2, 1, 1, 1, 26, 5, 6, 2, 10, 1, 3, 2, 4, 1, 8, 1, 3, 5, 21, 5, 2, 1, 1, 9, 1, 2, 1, 1, 1, 4, 2, 9, 1, 5, 18, 2, 2, 1, 1, 6, 1, 8, 1, 3, 3, 12]]

[30, [821289330402749, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 2, 10, 1, 2, 1, 1, 1, 3, 5, 1, 4, 1, 2, 2, 1, 2, 1, 2, 1, 6, 6, 2, 20, 11, 1, 1, 6, 1, 1, 1, 2, 2, 1, 1, 4, 1, 3, 5, 6, 1, 1, 1, 2, 4, 2, 1, 8, 1, 6, 1, 1, 137, 7, 1, 3, 5, 1, 1, 2, 39, 3]]

[31, [2580156526864958, 1, 1, 23, 1, 1, 8, 12, 3, 4, 433, 1, 2, 13, 1, 1, 2, 1, 30, 2, 1, 1, 1, 5, 8, 11, 2, 1, 3, 7, 1, 9, 180, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 10, 2, 2, 1, 1, 1, 2, 2, 15, 649]]

[32, [8105800789910709, 1, 1, 1, 7, 1, 1, 3, 1, 8, 2, 2, 1, 3, 3, 2, 9, 2, 3, 99, 35, 5, 4, 3, 7, 1, 4, 1, 17, 1, 1, 1, 2, 2, 1, 1, 1, 10, 5, 70, 1, 40, 1, 2, 10, 7, 2, 2, 1, 1, 49, 1, 3, 16]]

[33, [25465124213045828, 2, 7, 1, 736, 3, 1, 5, 1, 35, 1, 12, 5, 3, 6, 3, 11, 2, 15, 1, 1, 2, 2, 2, 9, 2, 1, 4744, 4, 12, 3, 2, 15, 1, 1, 1, 200, 10, 1, 1, 3, 4, 6, 1, 2]]

[34, [80001047150456339, 1, 1, 4, 4, 1, 1, 16, 1, 1, 2, 2, 1, 1, 4, 37, 7, 1, 2, 1, 5, 1, 1, 2, 1, 24, 1, 8, 1, 1, 1, 1, 13, 2, 1, 13, 1, 3, 1, 3, 1, 2, 38, 3, 3, 1, 1, 2, 1, 2, 1, 4, 1, 1, 3, 2, 1, 124, 1, 11, 1, 2, 1, 12, 2, 2]]

[35, [251330702007364298, 1, 1, 1, 1, 1, 1, 7, 1, 3, 1, 1, 1, 5834, 1, 3, 34, 1, 3, 1, 2, 3, 1, 3, 2, 1, 2, 1, 7, 63, 3, 6, 3, 1, 6, 1, 12, 6, 1, 2, 2, 15, 1, 6, 4, 2, 5, 1, 10, 1, 5, 18, 13, 1, 4]]

[36, [789578687047901181, 11, 3, 1, 6, 37, 6, 1, 6, 1, 2, 3, 3, 2, 2, 1, 3, 10, 2, 2, 3, 2, 1, 1, 4, 90, 1, 2, 1, 3, 1, 1, 28, 40, 1, 1, 3, 2, 5, 1, 4, 37, 1, 12, 3, 9, 5, 1, 4, 1, 1, 2, 1, 1, 6]]

[37, [2480534602660760780, 2, 3, 3, 1, 2, 5, 1, 1, 1, 5, 1, 2, 1, 64, 7, 1, 1, 7, 1, 1, 16, 1, 84, 1, 2, 1, 1, 3, 4, 1228, 13, 2, 1, 13, 1, 1, 2, 4, 1, 1, 10, 4, 5, 2, 3, 1, 3, 1, 2, 2, 1, 2, 1, 2, 7, 2]]

[38, [7792829284694322855, 1, 1, 1, 1, 1, 7, 3, 3, 141, 1, 2, 2, 2, 1, 1, 192, 1, 5, 1, 1, 3, 1, 8, 14, 1, 2, 49, 1, 12, 1, 4, 2, 1, 12, 2, 14, 12, 246, 1, 35, 1, 1, 3, 6, 1, 1, 1, 1, 14, 4]]

[39, [24481895231475088054, 1, 1, 1, 3, 1, 3, 1, 1, 7, 1, 1, 3, 42, 1, 150, 1, 2, 1, 1, 5, 1, 1, 1, 8, 2, 9, 1, 2, 1, 2, 2, 15, 2, 1, 1, 5, 1, 1, 1, 1, 4, 4, 2, 2, 1, 17, 3, 2, 1, 1, 2, 5, 24, 2, 8, 2, 1, 1, 1, 7, 1, 2]]

[40, [76912142205157127257, 3, 1, 3, 2, 1, 2, 1, 4, 1, 7, 79, 2, 1, 3, 2, 2, 1, 1, 5, 7, 31, 1, 1, 4, 1, 1, 15, 1, 2, 1, 1, 18, 1, 8, 1, 7, 1, 3, 1, 1, 1, 2, 2, 1, 1, 3, 6, 4, 1, 7, 20, 1, 1, 3, 1, 1, 1, 3, 15, 2]]

[41, [241626620923575111130, 3, 2, 5, 4, 1, 2, 7, 1, 4, 8, 1, 18, 1, 5, 3, 1, 3, 1, 1, 2, 3, 4, 10, 1, 1, 10, 1, 2, 1, 4, 37, 1, 5, 1, 1, 2, 4, 1, 1, 1, 1, 1, 4, 1, 2, 3, 5, 1, 1, 110, 1, 1, 2, 3, 1, 4, 1, 9, 1, 1, 2]]

[42, [759092417205229390873, 1, 7, 11, 2, 1, 6, 2, 2, 1, 1, 2, 1, 2, 104, 25, 13, 6, 1, 3, 3, 2, 1, 7, 2, 2, 4, 2, 1, 5, 2, 3, 1, 1, 1, 1, 3, 1, 1, 1, 16, 5, 1, 18, 40, 2, 1, 2, 1, 1, 16, 4, 11]]

[43, [2384759161287667022584, 1, 16, 1, 2, 4, 1, 5, 1, 1, 12, 3, 1, 3, 178, 1, 23, 1, 5, 2, 1, 1, 4, 1, 1, 6, 1, 1, 2, 1, 4, 1, 14, 3, 1, 1, 10, 10, 1, 2, 12, 2, 1, 115, 1, 885, 1, 1, 2]]

[44, [7491941861682291566352, 1, 1, 4, 7, 5, 1, 1, 1, 5, 2, 1, 1, 5, 6, 1, 18, 2, 4, 139, 1, 25, 4, 2, 1, 1, 2, 86, 1, 3, 3, 1, 4, 3, 5, 1, 2, 10, 31, 2, 1, 3, 5, 14, 1, 2, 1, 1, 1, 1, 1, 1, 11]]

[45, [23536629513782926058032, 2, 23, 3, 4, 4, 2, 1, 2, 1, 1, 5, 1, 4, 2, 1, 1, 1, 1, 2, 2, 5, 1, 6, 2, 4, 2, 1, 5, 2, 30, 1, 1, 1, 3, 1, 39, 2, 5, 3, 4, 7, 1, 1, 40, 3, 12, 2, 1, 1, 2, 1, 8, 1, 1, 1, 15]]

[46, [73942502370765147684186, 1, 1, 2, 2, 8, 7, 1, 2, 2, 1, 16, 1, 1, 1, 1, 6, 1, 2, 1, 9, 20, 7, 1, 9, 10, 5, 9, 1, 1, 16, 1, 2, 1, 8, 1, 2, 1, 3, 10, 2, 1, 1, 1, 1, 4, 1, 1, 301, 1, 3, 1, 10]]

[47, [232297222236041657886385, 1, 3, 1, 77, 2, 13, 1, 1, 5, 3, 1, 3, 2, 24, 51, 25, 1, 1, 2, 6, 13, 4, 1, 1, 13, 3, 1, 1, 8, 1, 41, 3, 9, 13, 2, 7, 1, 1, 1, 14, 1, 1, 1, 5, 1, 4]]

[48, [729783246826064035206280, 1, 1, 12, 1, 2, 2, 2, 1, 10, 8, 2, 1, 1, 9, 5, 3, 8, 2, 1, 38, 3, 1, 4, 1, 20, 1, 1, 1, 16, 1, 2, 1, 7, 5, 1, 1, 10, 1, 1, 2, 6, 34, 4, 1, 1, 2, 3, 1, 2, 1, 13, 1, 1, 2, 2]]

[49, [2292681686941669566432911, 7, 4, 3, 1, 82, 5, 13, 3, 3, 1, 3, 1, 3, 1, 2, 1, 1, 5, 1, 11, 1, 110, 1, 11, 1, 1, 1, 3, 3, 1, 1, 1, 31, 1, 4, 1, 3, 7, 3, 14, 1, 2, 1, 4, 170, 3, 3]]

[50, [7202671944715803306364652, 1, 2, 19, 4, 1, 2, 1, 1, 116, 4, 1, 1, 1, 2, 2, 1, 7, 1, 3, 2, 8, 1, 2, 2, 9, 4, 9, 2, 3, 7, 1, 1, 1, 1, 7, 2, 1, 4, 11, 14, 1, 2, 2, 47, 3, 1, 1, 3, 2, 5]]

[51, [22627861267736477052126260, 8, 1, 4, 1, 1, 1, 3, 1, 1, 1, 215, 8, 170, 3, 2, 4, 1, 2, 7, 2, 1, 9, 1, 3, 1, 5, 2, 1, 3, 2, 3, 7, 1, 1, 1, 3, 3, 27, 1, 1, 1, 1, 9, 1, 2, 2, 1, 1, 1, 3, 2, 2, 1, 3]]

[52, [71087522725169941823644668, 1, 4, 1, 2, 1, 1, 33, 2, 2, 2, 1, 1, 1, 8, 1, 2, 1, 40, 4, 4, 2, 1, 1, 3, 28, 1, 1, 1, 26, 1, 1, 29, 3, 2, 150, 1, 7, 3, 104, 4, 4, 2, 3, 1, 19]]

[53, [223328039155291367651964249, 2, 1, 2, 20, 1, 10, 1, 3, 1, 27, 5, 1, 1, 1, 6, 2, 3, 1, 2, 7, 3, 1, 1, 1, 3, 9, 1, 1, 2, 2, 1, 3, 2, 2, 1, 1, 1, 5, 3, 3, 3, 258, 4, 7, 1, 1, 2, 2, 2, 5, 1, 5, 2]]

[54, [701605727150877054142644067, 33, 1, 1, 1, 2, 3, 4, 8, 21, 1, 4, 3, 2, 9, 271, 34, 1, 2, 1, 6, 2, 2, 6, 1, 2, 3, 1, 43, 61, 1, 1, 2, 1, 21, 1, 39, 2, 2, 3]]

[55, [2204159398133720289741067354, 1, 1, 1, 1, 2, 8, 2, 1, 7, 1, 2, 1, 5, 19, 10, 3, 2, 3, 101, 5, 1, 6, 1, 1, 5, 2, 22, 4, 17, 1, 4, 3, 7, 2, 2, 6, 1, 1, 2, 5, 3, 1, 3, 4, 2]]

[56, [6924570972517795883320726281, 1, 1, 4, 1, 2, 10, 5, 1, 1, 3, 1, 5, 5, 5, 2, 12, 29, 1, 5, 1, 70, 8, 1, 3, 1, 1, 4, 3, 10, 2, 2, 17, 1, 5, 1, 10, 4, 2, 3, 1, 3, 1, 1, 1, 1, 3, 3]]

[57, [21754181296523037597639725717, 1, 6, 1, 1, 4, 3, 4, 1, 2, 3, 2, 3, 22, 1, 2, 2, 1, 1, 2, 2, 1, 4, 15, 1, 63, 6, 5, 3, 2, 17, 5, 1, 29, 1, 2, 3, 2, 3, 1, 1, 1, 4, 1, 2, 1, 2, 10]]

[58, [68342776146017258398967520977, 1, 188, 1, 2, 2, 1, 2, 2, 4, 3, 8, 1, 3, 3, 2, 1, 2, 3, 2, 4, 1, 3, 14, 1, 1, 2, 2, 5, 1, 1, 1, 1, 3, 1, 5, 1, 13, 1, 1, 1, 3, 14, 3, 15, 1, 4, 2, 4, 1, 1, 1, 1, 2]]

[59, [214705163466259581466085919504, 117, 1, 2, 1, 3, 1, 3, 4, 3, 9, 2, 1, 2, 4, 1, 1, 7, 3, 1, 2, 1, 2, 4, 2, 1, 2, 45, 1, 92, 2, 4, 3, 114, 5, 1, 29, 1, 1, 1, 1, 3, 11]]

[60, [674516164233396768200126958915, 1, 1, 3, 1, 2, 1, 1, 1, 131, 1, 1, 1, 1, 1, 3, 3, 1, 1, 2, 8, 2, 1, 5, 1, 1, 17, 5, 10, 1, 3, 1, 1, 4, 2, 3, 1, 1, 1, 16, 1, 4, 2, 1, 2, 2, 2, 1, 3, 2, 2, 1, 13, 1, 25]]

[61, [2119055026283205737110078195749, 2, 36, 1, 1, 5, 1, 2, 1, 4, 2, 7, 1, 1, 9, 2, 2, 1, 71, 17, 1, 45, 4, 1, 1, 1, 26, 2, 3, 6, 3, 18, 2, 1, 3, 3, 1, 7, 1, 2, 1, 4, 3]]

[62, [6657207703123845367424584915447, 1, 1, 7, 2, 1, 3, 1, 8, 34, 1, 18, 3, 4, 2, 1, 1, 6, 3, 2, 1, 389, 1, 2, 21, 4, 7, 9, 1, 6, 1, 1, 1, 6, 1, 5, 6, 77]]

[63, [20914234813555253845753030071176, 1, 4, 1, 1, 1, 1, 1, 2, 2, 2, 1, 5, 1, 2, 2, 1, 1, 1, 1, 1, 71, 20, 1, 1, 1, 1, 6, 1, 1, 2, 1, 18, 6, 1, 10, 1, 6, 4, 1, 1, 2, 1, 52, 3, 3, 2, 5, 2, 7, 1, 2]]

[64, [65704006445717084572022626334540, 1, 31, 1, 1, 11, 1, 5, 11, 1, 47, 1, 5, 1, 1, 2, 28, 10, 1, 3, 2, 7, 1, 5, 1, 6, 5, 1, 11, 5, 3, 1, 5, 2, 1, 2, 2, 5, 2, 2, 1, 2, 1, 13]]

[65, [206415223961281214949635806621076, 8, 1, 2, 15, 2, 1, 3, 1, 2, 2, 1, 1, 1, 7, 1, 1, 7, 3, 3, 1, 1, 1, 5, 1, 20, 1, 3, 392, 1, 1, 2, 3, 2, 1, 4, 2, 1, 2, 17, 1, 25, 1, 6, 7]]

[66, [648472551185852924760805487074064, 1, 1, 1, 3, 1, 1, 1, 2, 4, 1, 1, 33, 1, 46, 1, 2, 3, 1, 1, 1, 4, 3, 2, 3, 4, 3, 1, 35, 1, 1, 1, 3, 22, 1, 3, 2, 14, 19, 1, 1, 1, 12, 8, 4]]

[67, [2037236602860106711985144771884927, 2, 1, 1, 5, 2, 1, 1, 1, 1, 1, 13, 1, 24, 1, 1, 1, 1, 5, 1, 4, 3, 2, 1, 1, 19, 1, 1, 2, 2, 1, 14, 1, 6, 2, 1, 2, 6, 1, 3, 2, 1, 6, 4, 3, 1, 7, 16, 3, 1, 1, 2]]

[68, [6400167545169538406684014293670341, 1, 1, 4, 10, 1, 6, 1, 1, 3, 1, 1, 4, 5, 1, 18, 1, 3, 1, 5, 1, 2, 7, 2, 4, 1, 1, 1, 1, 2, 6, 1, 2, 2, 1, 6, 4, 1, 1, 1, 18, 1, 4, 1, 4, 1, 3, 1, 1, 1, 1, 1, 1, 17]]

[69, [20106719341648443041008873880038160, 1, 10, 11, 1, 54, 1, 2, 57, 2, 2, 1, 1, 10, 1, 1, 4, 9, 1, 1, 1, 1, 1, 1, 5, 1, 5, 4, 1, 5, 7, 6, 50, 1, 24, 1, 2, 6]]

[70, [63167121771514552681892760031018334, 1, 2, 1, 1, 3, 1, 1, 1, 3, 2, 1, 2, 3, 1, 2, 4, 51, 2, 2, 1, 2, 2, 1, 4, 1, 3, 2, 7, 3, 24, 539, 8, 1, 2, 2, 9, 1, 1, 5, 1, 1, 10]]

[71, [198445365705802004702001456166116205, 1, 3, 2, 34, 1, 4, 1, 5, 1, 1, 1, 10, 1, 1, 1, 1, 1, 10, 7, 36, 3, 6, 3, 1, 1, 2, 4, 4, 2, 1, 4, 11, 294, 1, 56, 2]]

[72, [623434503040287472342007784213127357, 1, 3, 1, 1, 2, 1, 6, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 2, 1, 5, 3, 4, 1, 1, 7, 44, 1, 1, 2, 13, 1, 97, 1, 3, 5, 1, 1, 1, 2, 1, 4, 2]]

[73, [1958577254745770740635072198655932631, 27, 1, 1, 1, 4, 1, 4, 35, 7, 1, 19, 1, 1, 5, 2, 4, 3, 1, 2, 1, 1, 1, 1, 1, 3, 2, 1, 5, 8, 1, 1, 1, 1, 2, 41, 1, 1, 1, 48, 1, 19]]

[74, [6153051914997378363457293172896716008, 1, 18, 1, 4, 23, 2, 1, 15, 1, 2, 14, 1, 4, 3, 1, 13, 1, 3, 1, 1, 3, 4, 6, 1, 4, 2, 4, 2, 7, 4, 4, 5, 2, 9, 3, 2]]

[75, [19330382693312372796273669213182810875, 1, 3, 6, 1, 4, 2, 2, 11, 1, 2, 1, 1, 1, 1, 2, 3, 33, 1, 1, 1, 2, 2, 4, 1, 10, 1, 1, 5, 47, 13, 5987]]

[76, [60728188260389431620181363923786356434, 5, 16, 4, 3, 1, 3, 1, 4, 6, 1, 4, 12, 1, 5, 2, 2, 4, 1, 1, 7, 3, 1, 3, 3, 1, 1, 1, 2, 13, 1, 1, 1, 2, 1, 2, 1, 36, 9, 2, 4]]

[77, [190783230104657364116871823923953977404, 4, 4, 4, 15, 2, 2, 4, 1, 5, 1, 24, 1, 6, 2, 3, 3, 1, 11, 1, 4, 2, 2, 1, 3, 1, 2, 2, 1, 10, 1, 3, 6, 2, 36, 2, 4, 3]]

[78, [599363194124922655319699900093848879647, 1, 17, 1, 13, 1, 2, 6, 5, 4, 1, 8, 1, 1, 42, 2, 31, 1, 26, 7, 5, 1, 2, 1, 2, 1, 20, 1, 23, 5, 1, 5, 4]]

[79, [1882955007494970137423872697623397063368, 1, 8, 3, 2, 1, 3, 1, 2, 6, 2, 1, 6, 3, 3, 4, 1, 1, 1, 1, 3, 5, 2, 2, 2, 1, 4, 1, 4, 1, 1, 1, 1, 1, 18, 1, 6, 1, 3, 4, 1, 76]]

[80, [5915477618586312249935148194956040327282, 1, 2, 4, 1, 3, 3, 10, 1, 1, 12, 4, 7, 1, 25, 4, 10, 1, 1, 1, 3, 2, 2, 1, 3, 1, 1, 1, 4, 3, 7, 1, 1, 1, 3, 1, 1, 1, 1, 27]]

[81, [18584021029025603512479467959285372526679, 270, 5, 3, 10, 4, 1, 2, 3, 1, 5, 1, 3, 1, 18, 2, 1, 1, 9, 1, 8, 953, 8, 3, 1, 8, 4]]

[82, [58383423938945065690063367398575102156630, 1, 1, 1, 50, 19, 6, 4, 1, 10, 1, 3, 1, 11, 2, 14, 5, 7, 6, 1, 1, 12, 1, 3, 1, 1, 10, 1, 1, 1, 84]]

[83, [183416935738008287619993279499823507509433, 1, 20, 3, 2, 6, 1, 4, 5, 3, 1, 1, 7, 1, 2, 1, 1, 1, 1, 24, 1, 1, 3, 39, 1, 8, 2, 4, 3, 6, 13, 1, 1, 10]]

[84, [576221297858478037757896202112843514258698, 1, 6, 3, 3, 3, 2, 2, 5, 5, 3, 1, 4, 1, 3, 1, 2, 2, 1, 1, 1, 1, 4, 38, 2, 2, 1, 110, 5, 1, 3, 1, 2, 1, 2, 2]]

[85, [1810252596194170662517162499694380571752661, 5, 2, 3, 1, 7, 1, 2, 1, 2, 1, 4, 1, 2, 2, 4, 1, 5, 2, 4, 1, 3, 1, 50, 1, 3, 48, 1, 7, 2, 1, 1, 1, 6, 1, 4, 5]]

[86, [5687076257345457055936827972911358112818998, 3, 59, 1, 9, 1, 2, 5, 1, 2, 1, 4, 1, 16, 1, 1, 1, 2, 1, 16, 1, 21, 1, 3, 1, 5, 1, 1, 1, 3, 1, 1, 16, 3, 1, 6]]

[87, [17866476990481424293061935394100969629693609, 1, 1, 1, 2, 1, 3, 2, 1, 13, 1, 6, 17, 13, 3, 9, 317, 1, 2, 1, 4, 1, 13, 1, 1, 4, 2, 5, 1, 1, 1, 1, 2]]

[88, [56129192858827520816193436882886842322337670, 25, 11, 2, 1, 2, 12, 9, 7, 1, 95, 99, 3, 1, 4, 228, 1, 5, 6, 5]]

[89, [176335059937217224018860024808747332179340048, 2, 51, 1, 6, 48, 1, 16, 1, 1, 1, 1, 2, 12, 1, 7, 3, 1, 72, 1, 1, 1, 2, 2, 1, 3, 19, 2, 1, 4, 1, 2]]

[90, [553972928869077498297581459358161277972163803, 10, 2, 1, 3, 8, 3, 2, 6, 17, 1, 17, 1, 1, 4, 2, 1, 1673, 1, 1, 1, 2, 2, 2, 1, 79, 3]]

[91, [1740357283622714955277630726239733032237686240, 9, 2, 2, 7, 3, 8, 6, 1, 8, 2, 3, 7, 2, 1, 2, 4, 5, 1, 1, 1, 39, 7, 1, 1, 1, 75, 2]]

[92, [5467493656850609485803318470436356848616987732, 1, 1, 1, 1, 4, 13, 22, 1, 1, 8, 1, 1, 3, 1, 2, 2, 1, 18, 12, 1, 2, 1, 5, 3, 1, 3, 1, 290, 8]]

[93, [17176637905910668669319694724112097956068639929, 16, 2, 6, 1, 14, 1, 12, 1, 6, 3, 6, 4, 1, 1, 1, 2, 1, 19, 1, 30, 1, 1, 1, 98, 1, 1, 1, 3, 2]]

[94, [53961999458580926862279289940854110180185778687, 1, 1, 1, 5, 2, 2, 3, 14, 2, 1, 12, 1, 31, 3, 3, 2, 2, 23, 1, 16, 1, 1, 8, 1, 3, 1, 2, 1, 5]]

[95, [169526621072094240051062813849433542430041251045, 1, 3, 1, 1, 1, 1, 1, 3, 1, 6, 2, 7, 1, 5, 1, 1, 35, 5, 1, 19, 12, 1, 2, 1, 4, 2, 33, 1, 1, 12]]

[96, [532583587347991902715777143640488633081518781241, 1, 2, 1, 1, 1, 2, 5, 1, 1, 3, 1, 2, 13, 1, 1, 1, 1, 6, 1, 52, 1, 2, 9, 3, 1, 1, 1, 16, 1, 1, 3, 1, 5, 10]]

[97, [1673160685434949314609400471476613579121899911697, 1, 11, 1, 1, 1, 1, 1, 40, 1, 1, 1, 4, 1, 2, 1, 14, 3, 3, 2, 1, 4, 1, 1, 9, 1, 36, 1, 2, 5, 1, 1, 4]]

[98, [5256389317637699735335333000834901305449119091054, 1, 2, 1, 19, 8, 1, 8, 1, 3, 2, 1, 2, 1, 2, 9, 1, 5, 12, 4, 1, 1, 2, 7, 1, 8, 1, 3, 3, 1, 3]]

[99, [16513434064698463682619509822252680822186405912861, 3, 3, 2, 9, 1, 1, 1, 6, 3, 1, 21, 1, 1, 24, 1, 68, 1, 11, 2, 36, 3, 1, 1, 2]]

[100, [51878483143196131920862615246303013562686760680405, 1, 3, 1, 1, 1, 6, 2, 2, 18, 1, 5, 11, 2, 1, 2, 1, 1, 3, 2, 16, 3, 1, 4, 3, 1, 3, 2, 4, 2]]

240325

PARI


シュワルツの定理

「ラマヌジャンの遺した関数」の2.7 に載っている定理を確かめてみる。

a(n, m) = binomial(n^2, n)%n^m;
for(m=1, 6, for(n=1, 50, print1(a(n, m),", ")); print)

出力結果

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 

0, 2, 3, 12, 5, 12, 7, 56, 9, 40, 11, 108, 13, 84, 90, 240, 17, 144, 19, 80, 315, 220, 23, 72, 25, 312, 27, 560, 29, 0, 31, 992, 759, 544, 770, 720, 37, 684, 1053, 520, 41, 252, 43, 1408, 1125, 1012, 47, 1872, 49, 1200, 

0, 6, 3, 28, 5, 120, 7, 120, 9, 440, 11, 828, 13, 1064, 315, 496, 17, 2088, 19, 2080, 7812, 3608, 23, 648, 25, 5720, 6588, 10752, 29, 1800, 31, 18400, 29073, 12104, 36295, 12384, 37, 16568, 36036, 54920, 41, 12600, 43, 69168, 74025, 28520, 47, 54864, 49, 36200, 

0, 6, 3, 28, 5, 1200, 7, 3192, 2925, 6440, 11, 828, 13, 14784, 10440, 12784, 17, 25416, 19, 34080, 118944, 205920, 23, 111240, 156275, 286936, 262467, 515648, 29, 244800, 31, 575456, 640002, 169320, 36295, 712224, 37, 1827344, 2230839, 2166920, 41, 2012976, 43, 1091376, 3445650, 1683232, 47, 54864, 823592, 2036200, 

0, 6, 84, 796, 5, 3792, 7, 27768, 22608, 56440, 11, 83772, 13, 168448, 61065, 471536, 17, 445320, 19, 994080, 2258235, 4188272, 23, 6083208, 156275, 743912, 13548492, 3588928, 29, 16444800, 31, 28887008, 13685133, 2841992, 42053795, 49421088, 37, 24763840, 9171162, 45686920, 41, 101587248, 43, 98541872, 15747525, 64367616, 47, 153998928, 58471602, 133286200, 

0, 6, 84, 1820, 6255, 34896, 16814, 126072, 258804, 456440, 805266, 2820924, 2227771, 3933216, 1579815, 7811568, 9939016, 26899272, 12380514, 994080, 47183346, 112414544, 32181738, 22008456, 205234400, 131439048, 157037562, 365006656, 184600370, 210844800, 429437296, 699975648, 796392993, 502631656, 1460144420, 2044804896, 901471478, 1767937536, 2264776137, 1888886920, 2780548865, 1016425872, 735042258, 6860107056, 6843288150, 9332701536, 1834760103, 3211646544, 9380154819, 8883286200,