A323363 Dirichlet inverse of Dedekind's psi, A001615.

1, -3, -4, 3, -6, 12, -8, -3, 4, 18, -12, -12, -14, 24, 24, 3, -18, -12, -20, -18, 32, 36, -24, 12, 6, 42, -4, -24, -30, -72, -32, -3, 48, 54, 48, 12, -38, 60, 56, 18, -42, -96, -44, -36, -24, 72, -48, -12, 8, -18, 72, -42, -54, 12, 72, 24, 80, 90, -60, 72, -62, 96, -32, 3, 84, -144, -68, -54, 96, -144, -72, -12, -74, 114, -24

Offset

1,2

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Formula

G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} psi(k) * A(x^k). - Ilya Gutkovskiy, Sep 04 2019

From Amiram Eldar, Oct 14 2020: (Start)

Multiplicative with a(p^e) = (-1)^e * (p+1).

a(n) = A008836(n) * A048250(n). (End)

Dirichlet g.f.: zeta(2*s)/(zeta(s-1)*zeta(s)). - Amiram Eldar, Dec 05 2022

Mathematica

psi[n_] := If[n == 1, 1, n Times @@ (1 + 1/FactorInteger[n][[All, 1]])];
a[n_] := a[n] = If[n == 1, 1, -Sum[psi[n/d] a[d], {d, Most@ Divisors[n]}]];
Array[a, 75] (* Jean-François Alcover, Feb 15 2020 *)
f[p_, e_] := (-1)^e * (p + 1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 14 2020 *)

Prog

(PARI)
A001615(n) = (n * sumdivmult(n, d, issquarefree(d)/d)); \\ From A001615
A323363(n) = if(1==n, 1, -sumdiv(n, d, if(d<n, A001615(n/d)*A323363(d), 0)));

Crossrefs

Cf. A048250 (absolute values).

Cf. A001615, A008836, A323364, A323372.

Sequence in context: A238162 A367503 A048250 * A073181 A183100 A340323

Adjacent sequences: A323360 A323361 A323362 * A323364 A323365 A323366

Keyword

sign,mult,easy

Author

Antti Karttunen, Jan 13 2019

Status

approved