A351654 Dirichlet g.f.: zeta(s) / (zeta(s-1) * zeta(s-2)).

1, -5, -11, 3, -29, 55, -55, 3, 16, 145, -131, -33, -181, 275, 319, 3, -305, -80, -379, -87, 605, 655, -551, -33, 96, 905, 16, -165, -869, -1595, -991, 3, 1441, 1525, 1595, 48, -1405, 1895, 1991, -87, -1721, -3025, -1891, -393, -464, 2755, -2255, -33, 288, -480, 3355, -543, -2861, -80, 3799

Offset: 1

Author

Ilya Gutkovskiy, Feb 16 2022

Keywords

• sign
• mult

Comments

Dirichlet inverse of A069097.

Links

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

Crossrefs

Cf. A000005, A023900, A046970, A055615, A069097, A101035, A328254, A334657.

Programs

Mathematica

A069097[n_] := Sum[GCD[n, k]^2, {k, 1, n}]; a[1] = 1; a[n_] := a[n] = -Sum[A069097[n/d] a[d], {d, Most @ Divisors[n]}]; Table[a[n], {n, 1, 55}]
f[p_, e_] := If[e == 1, 0, p^3] - p^2 - p + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 25 2025 *)

PARI

up_to = 20000;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA069097(n) = sumdiv(n, d, moebius(n/d)*d*sigma(d));
v351654 = DirInverseCorrect(vector(up_to, n, A069097(n)));
A351654(n) = v351654[n]; \\ Antti Karttunen, Feb 16 2022

PARI

for(n=1, 100, print1(direuler(p=2, n, (1 - p*X)*(1 - p^2*X)/(1 - X))[n], ", ")) \\ Vaclav Kotesovec, Feb 16 2022

Formula

a(1) = 1; a(n) = -Sum_{d|n, d < n} A069097(n/d) * a(d).

a(n) = Sum_{d|n} A023900(n/d) * A334657(d).

a(n) = Sum_{d|n} A046970(n/d) * A055615(d).

a(n) = Sum_{d|n} A000005(n/d) * A328254(d).

Multiplicative with a(p) = -p^2 - p + 1, and a(p^e) = p^3 - p^2 - p + 1 for e >= 2. - Amiram Eldar, May 25 2025