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

1,2

Comments

Dirichlet inverse of A069097.

Links

Table of n, a(n) for n=1..55.

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).

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}]

Prog

(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(d<n, v[n/d]*u[d], 0)))); (u) }; \\ Compute the Dirichlet inverse of the sequence given in input vector v.
A069097(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

Crossrefs

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

Sequence in context: A098147 A100298 A066461 * A117069 A145355 A351338

Adjacent sequences: A351651 A351652 A351653 * A351655 A351656 A351657

Keyword

sign,mult

Author

Ilya Gutkovskiy, Feb 16 2022

Status

approved