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

Offset: 1

Author

Zhining Yang, Oct 28 2024

Keywords

• nonn
• easy

Comments

a(n) = 0 for primes n not less than 23.

Examples

			For n=5, a(n) = 8 because 5*x*y*z = (x + 5)*(y + 5)*(z + 5), 0 < x <= y <= z has 8 integer solutions: {{2,12,595}, {2,14,95}, {2,15,70}, {2,20,35}, {3,6,220}, {3,10,20}, {4,5,45}, {5,5,20}}.
		

Crossrefs

Cf. A374059.

Programs

Mathematica

Table[{n, Length@Solve[n*x*y*z == (x + n) (y + n) (z + n) && 0 < x <= y <= z, {x, y, z}, Integers]}, {n, 60}][[All, 2]]