求一类方程的正整数解的数量

northwolves

设\(a(r,n)\)为方程 $r*\prod_{k=1}^nC_k=\prod_{k=1}^n(r+C_k)$ 的正整数解的数量，其中\(C_k<=C_{k+1}\)

$r>5$时计算量太大，难以继续了。

附目前计算结果：
$a(2,n)={0, 1, 2, 20, 374, 21313, 5115140...}$,OEIS搜索到序列
A263207
Number of integer solutions for Product_{k=1..n}(c(k) + 1) = 2 * Product_{k=1..n}(c(k) - 1) with 1 < c(k) <= c(k+1).

0, 1, 2, 20, 374, 21313, 5115140

$a(3,n)={0, 2,17, 450,35472,12127741 ...}$,OEIS未搜索到。

$a(4,n)={0, 2,15, 375,289010...}$

$a(5,n)={0, 0, 8, 301,36366...}$

$a(6,n)={0, 2, 18, 478,35534...}$

$a(7,n)={0, 0, 4, 228,33048...}$

$a(8,n)={0, 0, 11, 359, 55548...}$

$a(9,n)={0, 0, 5, 238,30891...}$

$r=3,r=4,r=5$相关序列：

A375787
a(n) is the number of solutions of n*x*y*z = (x + n)*(y + n)*(z + n), 0 < x <= y <= z.

0, 20, 17, 15, 8, 18, 4, 11, 5, 13, 1, 22, 2, 10, 13, 4, 1, 15, 1, 15, 9, 6, 0, 17, 3, 0, 1, 8, 0, 24, 0, 1, 6, 2, 6, 13, 0, 0, 4, 11, 0, 21, 0, 4, 10, 0, 0, 7, 0, 3, 2, 4, 0, 4, 1, 5, 0, 0, 0, 29


A380749
a(n) is the number of positive integer solutions of n*x*y*z*w = (x + n) * (y + n) * (z + n) * (w + n), x <= y <= z <= w.

0, 374, 450, 375, 301, 478, 228, 359, 238, 515, 206, 879, 259, 506, 780, 349, 284, 762, 135, 916, 905, 493, 99, 1189, 423, 306, 318, 869, 70, 1879, 97, 311, 714, 250, 778, 1300, 109, 258, 483, 1334, 71, 1987, 93, 545, 1451, 303, 64, 1156, 202, 504, 481, 822, 71


A381644
a(n) is the number of positive integer solutions of n*x*y*z*v*w = (x + n) * (y + n) * (z + n) * (v + n) * (w + n), x <= y <= z <= v<= w.

0, 21313, 35472, 28901, 36366, 35534, 33048, 55548, 30891, 60741, 76106, 161909, 88494, 114437, 220621, 76856, 56832, 195942, 33510, 212618, 222606, 154046, 21700, 324700, 107022, 94149, 109693, 244884, 35992, 592482, 39051, 134282, 213723, 104829, 363935, 355519, 70334, 110560, 158300, 485946, 46982, 650655

northwolves

a(3,6)=12127741 计算了五天才得到结果。由于计算中断过一次电，可能有误差，仅供参考。

northwolves

其他相关数列：

A374059
a(n) is the smallest integer k such that k*x*y*z = (x + k) * (y + k) * (z + k), 0 < x <= y <= z has exactly n integer solutions.
1, 11, 13, 25, 7, 9, 22, 48, 5, 21, 14, 8, 280, 10, 1020, 4, 70, 3, 6, 240, 2, 42, 12, 660, 30

A380750
a(n) is the smallest integer k such that k*x*y*z*w = (x + k) * (y + k) * (z + k) * (w + k), 0 < x <= y <= z <= w has exactly n integer solutions.

1019, 1559, 1637, 1103, 743, 419, 1039, 359, 311, 479, 653, 509, 389, 251, 593, 521, 263, 197, 1061, 131, 353, 269, 239, 167, 89, 179, 337, 113, 139, 83, 181, 229, 934, 898, 277, 151, 103, 554, 1042, 281, 109, 107, 566, 283, 1299, 79, 386, 157, 1959, 173, 241, 71, 59, 101, 1018, 326, 878, 993, 1366, 97, 127, 254, 1418, 47

wayne

northwolves 发表于 2025-3-9 07:21
a(3,6)=12127741 计算了五天才得到结果。由于计算中断过一次电，可能有误差，仅供参考。 ...

是Solve硬算的吗，比如这种

1. Block[{r=3,n=4,param},param=c/@Range[n];Length@Solve[r Times@@param==Times@@(r+param)&&LessEqual@@Flatten[{0,param}],param,Integers]]

复制代码