A343491 Number of representations of n! as a sum of 3 tetrahedral numbers (A000292).

1, 1, 1, 2, 2, 1, 2, 1, 3, 5, 2, 3, 6, 5, 8, 8, 7, 2, 7, 8, 3, 11, 2, 2

Offset: 1

Author

Altug Alkan, Apr 17 2021

Keywords

• nonn
• more

Comments

Conjecture I: There are infinitely many n such that a(n) >= 1.

Conjecture II: Natural density of numbers n such that a(n) >= 1 is 1.

Conjecture III: Numbers n such that a(n) = 0 is a finite sequence.

Conjecture IV: a(n) >= 1 for all n.

See Links section for some solutions.

Examples

			a(4) = 2 because 4! = 0 + 4 + 20 = 4 + 10 + 10.
a(24) = 2 because 24! = f(11393630) + f(118661018) + f(127041924) = f(81298034) + f(61098204) + f(143537134) where f = A000292.
		

Links

• Altug Alkan, A note on sequence

Crossrefs

Cf. A000142, A000292, A102796, A104246, A102799, A267414, A308852, A341794.

Programs

Mathematica

Table[Length[Solve[{i*(i + 1)*(i + 2) + j*(j + 1)*(j + 2) + k*(k + 1)*(k + 2) == 6*n!, i >= 0, j >= 0, k >= 0, i <= j, j <= k, k < (6*n!)^(1/3)}, Integers]], {n, 1, 10}] (* Vaclav Kotesovec, Apr 19 2021 *)