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A375576
a(n) is the number of partitions n = x + y + z of positive integers such that x*y*z is a perfect square.
3
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
0,12
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.
LINKS
Felix Huber, Maple Codes
EXAMPLE
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.
MAPLE
See Huber link.
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Felix Huber, Aug 19 2024
STATUS
approved