A375580 a(n) is the number of partitions n = x + y + z of positive integers such that x*y*z is a perfect cube.

0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 3, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 0, 2, 2, 2, 2, 1, 2, 3, 2, 2, 3, 2, 0, 1, 1, 3, 1, 3, 2, 2, 1, 1, 1, 2, 1, 4, 1, 2, 3, 3, 3, 3, 1, 1, 4, 2, 2, 2, 3, 1, 2, 3, 1, 3, 4, 1, 3, 2, 2, 1, 2, 2, 3, 3, 2, 4

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 cube.

Conjecture: for n > 176, a(n) > 0. - Charles R Greathouse IV, Aug 20 2024

Examples

			a(21) = 3 because the three partitions [1, 4, 16], [3, 6, 12], [7, 7, 7] satisfy the conditions: 1 + 4 + 16 = 21 and 1*4*16 = 4^3, 3 + 6 + 12 = 21 and 3*6*12 = 6^3, 7 + 7 + 7 = 21 and 7*7*7 = 7^3.
See also linked Maple code.
		

Links

• Felix Huber and Charles R Greathouse IV, Table of n, a(n) for n = 0..10000 (up to 1000 from Huber)
• Felix Huber, Maple codes
• David A. Corneth, PARI program

Crossrefs

Cf. A338939, A375512, A375576, A103277, A069905.

Programs

Maple

# See Huber link.

PARI

a(n)=sum(x=1,n\3, sum(y=x,(n-x)\2, ispower(x*y*(n-x-y),3))) \\ Charles R Greathouse IV, Aug 20 2024

PARI

\\ See Corneth link

Python

from sympy import integer_nthroot
def A375580(n): return sum(1 for x in range(n//3) for y in range(x,n-x-1>>1) if integer_nthroot((n-x-y-2)*(x+1)*(y+1),3)[1]) # Chai Wah Wu, Aug 21 2024

Formula

Trivial upper bound: a(n) <= A069905(n). - Charles R Greathouse IV, Aug 23 2024