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A334988 Sum of tetrahedral numbers dividing n.

%I A334988 #11 Feb 16 2025 08:34:00
%S A334988 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,
%T A334988 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,
%U A334988 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
%N A334988 Sum of tetrahedral numbers dividing n.
%H A334988 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TetrahedralNumber.html">Tetrahedral Number</a>
%F A334988 G.f.: Sum_{k>=1} binomial(k+2,3) * x^binomial(k+2,3) / (1 - x^binomial(k+2,3)).
%F A334988 L.g.f.: log(G(x)), where G(x) is the g.f. for A068980.
%F A334988 a(n) = Sum_{d|n} A023533(d) * d.
%t A334988 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
%t A334988 nmax = 90; CoefficientList[Series[Log[Product[1/(1 - x^Binomial[k + 2, 3]), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax] // Rest
%o A334988 (PARI) ist(n) = my(k=sqrtnint(6*n, 3)); k*(k+1)*(k+2)==6*n; \\ A000292
%o A334988 a(n) = sumdiv(n, d, if (ist(d), d)); \\ _Michel Marcus_, May 19 2020
%Y A334988 Cf. A000292, A023533, A068980, A185027, A279495, A334987.
%K A334988 nonn
%O A334988 1,4
%A A334988 _Ilya Gutkovskiy_, May 18 2020