谢谢 mathe! 谢谢 northwolves! 一并给出!
(2): $a^2-b^2=n^2$ 解的数目关系式,Table[Floor[DivisorSigma[0, GCD[((n - 1)/2)^2, ((n + 1)/2)^2] n^2]/2], {n, 1,130}]
{0, 0, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 1, 1, 4, 3, 1, 2, 1, 4, 4, 1, 1, 7, 2, 1, 3, 4, 1, 4, 1, 4, 4, 1, 4, 7, 1, 1, 4,7, 1, 4, 1, 4, 7, 1, 1, 10, 2, 2, 4, 4, 1, 3, 4, 7, 4, 1, 1, 13, 1, 1, 7, 5, 4, 4, 1, 4, 4, 4, 1, 12, 1, 1, 7, 4, 4, 4, 1, 10, 4, 1, 1, 13, 4, 1,
(3): $a^2-b^2=n^3$ 解的数目关系式,Table[Floor[DivisorSigma[0, GCD[((n - 1)/2)^2, ((n + 1)/2)^2] n^3]/2], {n, 1, 130}]
{0, 1, 2, 2, 2, 4, 2, 4, 3, 4, 2, 10, 2, 4, 8, 5, 2, 7, 2, 10, 8, 4, 2, 16, 3, 4, 5, 10, 2, 16, 2, 7, 8, 4, 8, 17, 2, 4, 8,16, 2, 16, 2, 10, 14, 4, 2, 22, 3, 7, 8, 10, 2, 10, 8, 16, 8, 4, 2, 40, 2, 4, 14, 8, 8, 16, 2, 10, 8, 16, 2, 28, 2, 4, 14, 10, 8, 16,
(4): $a^2-b^2=n^4$ 解的数目关系式,Table[Floor[DivisorSigma[0, GCD[((n - 1)/2)^2, ((n + 1)/2)^2] n^4]/2], {n, 1, 130}]
{0, 1, 2, 3, 2, 7, 2, 5, 4, 7, 2, 17, 2, 7, 12, 7, 2, 13, 2, 17, 12, 7, 2, 27, 4, 7, 6, 17, 2, 37, 2, 9, 12, 7, 12, 31, 2, 7, 12, 27, 2, 37, 2, 17, 22, 7, 2, 37, 4, 13, 12, 17, 2, 19, 12, 27, 12, 7, 2, 87, 2, 7, 22, 11, 12, 37, 2, 17, 12, 37, 2, 49, 2,
(5): $a^2-b^2=n^5$ 解的数目关系式,Table[Floor[DivisorSigma[0, GCD[((n - 1)/2)^2, ((n + 1)/2)^2] n^5]/2], {n, 1, 130}]
{0, 2, 3, 4, 3, 12, 3, 7, 5, 12, 3, 27, 3, 12, 18, 9, 3, 22, 3, 27, 18, 12, 3, 42, 5, 12, 8, 27, 3, 72, 3, 12, 18, 12, 18, 49, 3, 12, 18, 42, 3, 72, 3, 27, 33, 12, 3, 57, 5, 22, 18, 27, 3, 32, 18, 42, 18, 12, 3, 162, 3, 12, 33, 14, 18, 72, 3, 27, 18,
(6): $a^2-b^2=n^6$ 解的数目关系式,Table[Floor[DivisorSigma[0, GCD[((n - 1)/2)^2, ((n + 1)/2)^2] n^6]/2], {n, 1, 130}]
{0, 2, 3, 5, 3, 17, 3, 8, 6, 17, 3, 38, 3, 17, 24, 11, 3, 32, 3, 38, 24, 17, 3, 59, 6, 17, 9, 38, 3, 122, 3, 14, 24, 17, 24, 71, 3, 17, 24, 59, 3, 122, 3, 38, 45, 17, 3, 80, 6, 32, 24, 38, 3, 47, 24, 59, 24, 17, 3, 269, 3, 17, 45, 17, 24, 122, 3, 38,
(7): $a^2-b^2=n^7$ 解的数目关系式,Table[Floor[DivisorSigma[0, GCD[((n - 1)/2)^2, ((n + 1)/2)^2] n^7]/2], {n, 1, 130}]
{0, 3, 4, 6, 4, 24, 4, 10, 7, 24, 4, 52, 4, 24, 32, 13, 4, 45, 4, 52, 32, 24, 4, 80, 7, 24, 11, 52, 4, 192, 4, 17, 32, 24, 32, 97, 4, 24, 32, 80, 4, 192, 4, 52, 60, 24, 4, 108, 7, 45, 32, 52, 4, 66, 32, 80, 32, 24, 4, 416, 4, 24, 60, 20, 32, 192, 4,
(8): $a^2-b^2=n^8$ 解的数目关系式,Table[Floor[DivisorSigma[0, GCD[((n - 1)/2)^2, ((n + 1)/2)^2] n^8]/2], {n, 1, 130}]
{0, 3, 4, 7, 4, 31, 4, 11, 8, 31, 4, 67, 4, 31, 40, 15, 4, 59, 4, 67, 40, 31, 4, 103, 8, 31, 12, 67, 4, 283, 4, 19, 40, 31, 40, 127, 4, 31, 40, 103, 4, 283, 4, 67, 76, 31, 4, 139, 8, 59, 40, 67, 4, 87, 40, 103, 40, 31, 4, 607, 4, 31, 76, 23, 40, 283,
(9): $a^2-b^2=n^9$ 解的数目关系式,Table[Floor[DivisorSigma[0, GCD[((n - 1)/2)^2, ((n + 1)/2)^2] n^9]/2], {n, 1, 130}]
{0, 4, 5, 8, 5, 40, 5, 13, 9, 40, 5, 85, 5, 40, 50, 17, 5, 76, 5, 85, 50, 40, 5, 130, 9, 40, 14, 85, 5, 400, 5, 22, 50, 40, 50, 161, 5, 40, 50, 130, 5, 400, 5, 85, 95, 40, 5, 175, 9, 76, 50, 85, 5, 112, 50, 130, 50, 40, 5, 850, 5, 40, 95, 26, 50, 400, |