A375576 a(n) is the number of partitions n = x + y + z of positive integers such that x*y*z is a perfect square.

0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 4, 1, 2, 3, 2, 2, 4, 2, 4, 3, 4, 2, 5, 4, 2, 6, 5, 2, 8, 4, 8, 4, 4, 5, 10, 5, 3, 8, 7, 6, 12, 5, 6, 7, 6, 7, 11, 5, 6, 8, 12, 6, 11, 8, 11, 11, 6, 3, 22, 6, 12, 12, 8, 9, 13, 12, 7, 14, 14, 6, 18, 7, 7, 18, 13, 14, 13, 7, 19, 10

Offset: 0

Author

Felix Huber, Aug 19 2024

Keywords

• nonn

Comments

a(n) is also the number of distinct integer-sided cuboids with total edge length 4*n whose unit cubes can be grouped to a square cuboid with height 1.

Examples

			a(24) = 4 because the four partitions [2, 4, 18], [3, 9, 12], [4, 4, 16], [4, 10, 10] satisfy the conditions: 2 + 4 + 18 = 24 and 2*4*18 = 12^2, 3 + 9 + 12 = 24 and 3*9*12 = 18^2, 4 + 4 + 16 = 24 and 4*4*16 = 16^2, 4 + 10 + 10 = 24 and 4*10*10 = 20^2.
See also linked Maple code.
		

Links

• Felix Huber, Table of n, a(n) for n = 0..1000
• Felix Huber, Maple Codes

Crossrefs

Cf. A338939, A375512, A375580.

Programs

Maple

See Huber link.

Python

from sympy.ntheory.primetest import is_square
def A375567(n): return sum(1 for x in range(n//3) for y in range(x,n-x-1>>1) if is_square((n-x-y-2)*(x+1)*(y+1))) # Chai Wah Wu, Aug 22 2024