A334988 Sum of tetrahedral numbers dividing n.

1, 1, 1, 5, 1, 1, 1, 5, 1, 11, 1, 5, 1, 1, 1, 5, 1, 1, 1, 35, 1, 1, 1, 5, 1, 1, 1, 5, 1, 11, 1, 5, 1, 1, 36, 5, 1, 1, 1, 35, 1, 1, 1, 5, 1, 1, 1, 5, 1, 11, 1, 5, 1, 1, 1, 61, 1, 1, 1, 35, 1, 1, 1, 5, 1, 1, 1, 5, 1, 46, 1, 5, 1, 1, 1, 5, 1, 1, 1, 35, 1, 1, 1, 89, 1, 1, 1, 5, 1, 11

Offset: 1

Author

Ilya Gutkovskiy, May 18 2020

Keywords

• nonn

Links

• Eric Weisstein's World of Mathematics, Tetrahedral Number

Crossrefs

Cf. A000292, A023533, A068980, A185027, A279495, A334987.

Programs

Mathematica

nmax = 90; CoefficientList[Series[Sum[Binomial[k + 2, 3] x^Binomial[k + 2, 3]/(1 - x^Binomial[k + 2, 3]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
nmax = 90; CoefficientList[Series[Log[Product[1/(1 - x^Binomial[k + 2, 3]), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax] // Rest

PARI

ist(n) = my(k=sqrtnint(6*n, 3)); k*(k+1)*(k+2)==6*n; \\ A000292
a(n) = sumdiv(n, d, if (ist(d), d)); \\ Michel Marcus, May 19 2020

Formula

G.f.: Sum_{k>=1} binomial(k+2,3) * x^binomial(k+2,3) / (1 - x^binomial(k+2,3)).

L.g.f.: log(G(x)), where G(x) is the g.f. for A068980.

a(n) = Sum_{d|n} A023533(d) * d.