A334987 Sum of centered triangular numbers dividing n.

1, 1, 1, 5, 1, 1, 1, 5, 1, 11, 1, 5, 1, 1, 1, 5, 1, 1, 20, 15, 1, 1, 1, 5, 1, 1, 1, 5, 1, 11, 32, 5, 1, 1, 1, 5, 1, 20, 1, 15, 1, 1, 1, 5, 1, 47, 1, 5, 1, 11, 1, 5, 1, 1, 1, 5, 20, 1, 1, 15, 1, 32, 1, 69, 1, 1, 1, 5, 1, 11, 1, 5, 1, 1, 1, 24, 1, 1, 1, 15, 1, 1, 1, 5, 86, 1, 1, 5, 1, 11

Offset: 1

Author

Ilya Gutkovskiy, May 18 2020

Keywords

• nonn

Links

• Eric Weisstein's World of Mathematics, Centered Triangular Number

Crossrefs

Cf. A005448, A185027, A280950, A300409, A334988.

Programs

Mathematica

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

PARI

isc(n) = my(k=(2*n-2)/3, m); (n==1) || ((denominator(k)==1) && (m=sqrtint(k)) && (m*(m+1)==k));
a(n) = sumdiv(n, d, if (isc(d), d)); \\ Michel Marcus, May 19 2020

Formula

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

L.g.f.: log(G(x)), where G(x) is the g.f. for A280950.