A332793 a(1) = 1; a(n) = n * Sum_{d|n, d < n} (-1)^(n/d) * a(d) / d.

1, 2, -3, 8, -5, -6, -7, 32, 0, -10, -11, -24, -13, -14, 15, 128, -17, 0, -19, -40, 21, -22, -23, -96, 0, -26, 0, -56, -29, 30, -31, 512, 33, -34, 35, 0, -37, -38, 39, -160, -41, 42, -43, -88, 0, -46, -47, -384, 0, 0, 51, -104, -53, 0, 55, -224, 57, -58, -59, 120

Offset

1,2

Links

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

Formula

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

Dirichlet g.f.: 1 / (zeta(s-1) * (1 - 2^(2 - s))).

a(n) = Sum_{d|n} A327268(d).

Multiplicative with a(2^e) = 2^(2*e-1), and a(p^e) = -p if e=1 and 0 for e>1, for odd primes p. - Amiram Eldar, Dec 02 2020

Mathematica

a[1] = 1; a[n_] := n Sum[If[d < n, (-1)^(n/d) a[d]/d, 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 60}]
terms = 60; A[_] = 0; Do[A[x_] = x + Sum[(-1)^k k A[x^k], {k, 2, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x] // Rest
f[p_, e_] := If[p == 2, p^(2*e - 1), -p*Boole[e == 1]]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 02 2020 *)

Crossrefs

Cf. A002129, A038838 (positions of 0's), A055615, A067856, A327268, A361987.

Partial sums give A361982.

Dirichlet inverse of A181983.

Sequence in context: A355264 A157488 A188385 * A102631 A100782 A110340

Adjacent sequences: A332790 A332791 A332792 * A332794 A332795 A332796

Keyword

sign,mult

Author

Ilya Gutkovskiy, Feb 24 2020

Status

approved