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

Offset: 1

Author

Zhining Yang, Jan 31 2025

Keywords

• nonn

Examples

			For n=5, a(5) = 301 because 5*x*y*z*w = (x + 5)*(y + 5)*(z + 5)*(w + 5), 0 < x <= y <= z <= w has 301 positive integer solutions: {{2,12,596,357595}, {2,12,597,179095}, {2,12,598,119595}, ..., {6,7,9,220}, {6,10,11,20}, {7,9,10,20}}.
		

Links

• Zhining Yang, Table of n, a(n) for n = 1..2130

Crossrefs

Cf. A374059, A375787, A380750.

Programs

Mathematica

Table[Length@Solve[n*x*y*z*w == (x + n)  (y + n)  (z + n) (w + n) &&
    0 < x <= y <= z <= w, {x, y, z, w}, Integers], {n, 10}]