A307704 Expansion of (1/(1 - x)) * Sum_{k>=1} (-x)^k/(1 - (-x)^k).

-1, 1, -1, 2, 0, 4, 2, 6, 3, 7, 5, 11, 9, 13, 9, 14, 12, 18, 16, 22, 18, 22, 20, 28, 25, 29, 25, 31, 29, 37, 35, 41, 37, 41, 37, 46, 44, 48, 44, 52, 50, 58, 56, 62, 56, 60, 58, 68, 65, 71, 67, 73, 71, 79, 75, 83, 79, 83, 81, 93, 91, 95, 89, 96, 92, 100, 98, 104, 100, 108

Offset

1,4

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.

Formula

a(n) = Sum_{k=1..n} (-1)^k*A000005(k).

a(n) = n*log(n)/2 + (gamma - log(2) - 1/2)*n + O(n^(131/416 + eps)) (Tóth, 2017). - Amiram Eldar, Oct 14 2022

Mathematica

nmax = 70; Rest[CoefficientList[Series[1/(1 - x) Sum[(-x)^k/(1 - (-x)^k), {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Sum[(-1)^k DivisorSigma[0, k], {k, 1, n}], {n, 1, 70}]
Accumulate[Array[(-1)^#*DivisorSigma[0, #] &, 70]] (* Amiram Eldar, Oct 14 2022 *)

Crossrefs

Cf. A000005, A006218, A051950, A059851, A068762, A068773, A092405, A263084 (bisection).

Cf. A001620 (gamma), A002162.

Sequence in context: A334082 A346612 A352528 * A139716 A168232 A253136

Adjacent sequences: A307701 A307702 A307703 * A307705 A307706 A307707

Keyword

sign

Author

Ilya Gutkovskiy, Apr 22 2019

Status

approved