A318368 a(n) = Sum_{d|n} (-1)^(n/d+1)*d*2^(d-1).

1, 3, 13, 27, 81, 183, 449, 987, 2317, 5043, 11265, 24399, 53249, 114243, 245853, 523227, 1114113, 2357175, 4980737, 10480587, 22020557, 46126083, 96468993, 201302799, 419430481, 872361987, 1811941645, 3757981275, 7784628225, 16105886823, 33285996545, 68718951387, 141733932045

Offset: 1

Author

Ilya Gutkovskiy, Aug 24 2018

Keywords

• nonn

Crossrefs

Cf. A001787, A054599, A083413, A098407.

Programs

Mathematica

Table[Sum[(-1)^(n/d + 1) d 2^(d - 1), {d, Divisors[n]}], {n, 33}]
nmax = 33; Rest[CoefficientList[Series[Sum[k 2^(k - 1) x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 33; Rest[CoefficientList[Series[Log[Product[(1 + x^k)^(2^(k - 1)), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]

PARI

a(n) = sumdiv(n, d, (-1)^(n/d+1)*d*2^(d-1)); \\ Michel Marcus, Aug 25 2018

Formula

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

L.g.f.: log(Product_{k>=1} (1 + x^k)^(2^(k-1))) = Sum_{n>=1} a(n)*x^n/n.

a(n) ~ n * 2^(n-1). - Vaclav Kotesovec, Aug 25 2018