A343491 - OEIS

%I #71 Apr 19 2021 03:37:31

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

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

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

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

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

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

%C See Links section for some solutions.

%H Altug Alkan, <a href="/A343491/a343491_1.pdf">A note on sequence</a>

%e a(4) = 2 because 4! = 0 + 4 + 20 = 4 + 10 + 10.

%e a(24) = 2 because 24! = f(11393630) + f(118661018) + f(127041924) = f(81298034) + f(61098204) + f(143537134) where f = A000292.

%t 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 *)

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

%K nonn,more

%O 1,4

%A _Altug Alkan_, Apr 17 2021