A348223 a(n) = Sum_{d|n} (-1)^(sigma(d) - 1).

1, 2, 0, 3, 0, 0, 0, 4, 1, 0, 0, 0, 0, 0, -2, 5, 0, 2, 0, 0, -2, 0, 0, 0, 1, 0, 0, 0, 0, -4, 0, 6, -2, 0, -2, 3, 0, 0, -2, 0, 0, -4, 0, 0, -2, 0, 0, 0, 1, 2, -2, 0, 0, 0, -2, 0, -2, 0, 0, -6, 0, 0, -2, 7, -2, -4, 0, 0, -2, -4, 0, 4, 0, 0, -2, 0, -2, -4, 0, 0, 1, 0, 0, -6, -2, 0, -2, 0, 0, -4, -2, 0, -2, 0, -2, 0, 0, 2, -2, 3, 0, -4, 0, 0, -6

Offset

1,2

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Formula

If p is an odd prime, a(p) = 0.

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

From Bernard Schott, Oct 19 2021: (Start)

If p is even prime = 2, a(2^k) = k+1 for k >= 0.

If p is odd prime, a(p^even) = 1 and a(p^odd) = 0 (compare with formulas in A347992). (End)

Mathematica

a[n_] := DivisorSum[n, (-1)^(DivisorSigma[1, #] - 1) &]; Array[a, 100] (* Amiram Eldar, Oct 08 2021 *)

Prog

(PARI) a(n) = sumdiv(n, d, (-1)^(sigma(d)-1));
(PARI) my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, (-1)^(sigma(k)-1)*x^k/(1-x^k)))

Crossrefs

Cf. A000203, A347991, A347992.

Sequence in context: A085200 A080024 A378998 * A035199 A035229 A348019

Adjacent sequences: A348220 A348221 A348222 * A348224 A348225 A348226

Keyword

sign

Author

Seiichi Manyama, Oct 08 2021

Status

approved