A339615 Number of nonempty sets of distinct positive integers whose sum of cubes is a cube, the largest integer of a set is n.

1, 1, 1, 1, 2, 1, 1, 3, 1, 6, 5, 9, 10, 25, 32, 51, 97, 144, 244, 463, 767, 1062, 2005, 4177, 5716, 12101, 21526, 35306, 64629, 114871, 205337, 372317, 718410, 1226320, 2361112, 4308192, 7301384, 14615750, 26382095, 47631200, 91388286, 171931627, 302867194, 578843590, 1112232587

Offset: 1

Author

Ilya Gutkovskiy, Dec 10 2020

Keywords

• nonn

Examples

			a(13) = 10 sets: {13}, {2, 3, 8, 13}, {4, 8, 11, 12, 13}, {1, 2, 6, 7, 11, 13}, {2, 5, 7, 8, 12, 13}, {3, 4, 8, 10, 11, 12, 13}, {1, 2, 3, 4, 5, 7, 11, 13}, {2, 3, 4, 6, 7, 8, 9, 13}, {1, 2, 5, 6, 7, 8, 9, 10, 12, 13} and {2, 3, 5, 7, 8, 9, 10, 11, 12, 13}.
		

Crossrefs

Cf. A000578, A126111, A339612.

Programs

Python

from functools import lru_cache
def perf_cube(n): return round(n**(1/3))**3 ==n
@lru_cache(maxsize=None)
def b(n, soc, c):
  if n == 0:
    if perf_cube(soc): return 1
    return 0
  return b(n-1, soc, c) + b(n-1, soc+n*n*n, c+1)
a = lambda n: b(n-1, n*n*n, 1)
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Dec 10 2020

Extensions

a(24)-a(41) from Michael S. Branicky, Dec 10 2020

a(42)-a(45) from Alois P. Heinz, Dec 11 2020