A053822 Dirichlet inverse of sigma_2 function (A001157).

1, -5, -10, 4, -26, 50, -50, 0, 9, 130, -122, -40, -170, 250, 260, 0, -290, -45, -362, -104, 500, 610, -530, 0, 25, 850, 0, -200, -842, -1300, -962, 0, 1220, 1450, 1300, 36, -1370, 1810, 1700, 0, -1682, -2500, -1850, -488, -234, 2650, -2210, 0, 49, -125, 2900, -680

Offset

1,2

Comments

sigma_2(n) is the sum of the squares of the divisors of n (A001157).

References

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 39.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

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

Multiplicative with a(p^1) = -1-p^2, a(p^2) = p^2, a(p^e) = 0 for e>=3. - Mitch Harris, Jun 27 2005

a(n) = Sum_{d|n} mu(n/d)*mu(d)*d^2. - Ilya Gutkovskiy, Nov 06 2018

From Peter Bala, Jan 26 2024: (Start)

a(n) = Sum_{d divides n} d * (sigma(d))^(-1) * phi(n/d), where (sigma(n))^(-1) = A046692(n) denotes the Dirichlet inverse of sigma(n) = A000203(n).

a(n) = Sum_{d divides n} d^2 * (sigma_k(d))^(-1) * J_(k+2)(n/d) for k >= 0, where (sigma_k(n))^(-1) denotes the Dirichlet inverse of the divisor sum function sigma_k(n) and J_k(n) denotes the Jordan totient function. (End)

Maple

f1:= proc(p, e) if e = 1 then -1-p^2 elif e=2 then p^2 else 0 fi end proc:
f:= n -> mul(f1(t[1], t[2]), t=ifactors(n)[2]);
map(f, [$1..100]); # Robert Israel, Jan 29 2018

Mathematica

a[n_] := Sum[MoebiusMu[n/d] MoebiusMu[d] d^2, {d, Divisors[n]}];
Array[a, 100] (* Jean-François Alcover, Mar 05 2019, after Ilya Gutkovskiy *)
f[p_, e_] := If[e == 1, -p^2 - 1, If[e == 2, p^2, 0]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 16 2020 *)

Prog

(PARI) seq(n)={dirdiv(vector(n, n, n==1), vector(n, n, sigma(n, 2)))} \\ Andrew Howroyd, Aug 05 2018
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 - X)*(1 - p^2*X))[n], ", ")) \\ Vaclav Kotesovec, Sep 16 2020

Crossrefs

Dirichlet inverse of sigma_k(n): A007427 (k = 0), A046692 (k = 1), A053825 (k = 3), A053826 (k = 4), A178448 (k = 5).

Cf. A001157,.

Sequence in context: A295121 A066200 A357913 * A376487 A262922 A276652

Adjacent sequences: A053819 A053820 A053821 * A053823 A053824 A053825

Keyword

sign,mult,look

Author

N. J. A. Sloane, Apr 08 2000

Status

approved