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A375580
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a(n) is the number of partitions n = x + y + z of positive integers such that x*y*z is a perfect cube.
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3
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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
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OFFSET
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0,21
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COMMENTS
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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.
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LINKS
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FORMULA
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EXAMPLE
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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.
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MAPLE
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# See Huber link.
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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