login
A375580
a(n) is the number of partitions n = x + y + z of positive integers such that x*y*z is a perfect cube.
6
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
0,21
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
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
FORMULA
Trivial upper bound: a(n) <= A069905(n). - Charles R Greathouse IV, Aug 23 2024
EXAMPLE
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.
MAPLE
# See Huber link.
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Felix Huber, Aug 19 2024
STATUS
approved