A316359 a(n) is the number of solutions to the Diophantine equation i^3 + j^3 + k^3 = n^3, where 0 < i <= j <= k.

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

Offset: 1

Author

Arlu Genesis A. Padilla, Jun 30 2018

Keywords

• nonn

Comments

The first number to have a nonzero number of solutions is 6, which is 3^3 + 4^3 + 5^3 = 6^3. Its cube 216 has been called Plato's number in reference to this.

First occurrence of k=0,1,2...: 0, 6, 18, 54, 87, 108, 216, 174, 348, 396, 324, 696, 864, 492, etc. - Robert G. Wilson v, Jul 02 2018

Examples

			a(18)=2, because 18^3 = 9^3 + 12^3 + 15^3 = 2^3 + 12^3 + 16^3.
		

Links

• Arlu Genesis A. Padilla, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A046080.

Programs

Mathematica

Array[Count[PowersRepresentations[#^3, 3, 3], ?(FreeQ[Differences@ #, 0] &)] &, 105] (* _Michael De Vlieger, Jun 30 2018 *)

PARI

a(n) = sum(i=1, n, sum(j=1, i, sum(k=1, j, i^3 + j^3 + k^3 == n^3))); \\ Michel Marcus, Jul 02 2018

PARI

a(n)={sum(i=1, n, sum(j=1, i, my(k); ispower(n^3-j^3-i^3, 3, &k) && k>=1 && k<=j ))} \\ Andrew Howroyd, Jul 07 2018

Python

from sympy.solvers.diophantine.diophantine import power_representation
def A316359(n): return len(list(power_representation(n**3,3,3))) # Chai Wah Wu, Nov 19 2024