A046692 Dirichlet inverse of sigma function (A000203).

1, -3, -4, 2, -6, 12, -8, 0, 3, 18, -12, -8, -14, 24, 24, 0, -18, -9, -20, -12, 32, 36, -24, 0, 5, 42, 0, -16, -30, -72, -32, 0, 48, 54, 48, 6, -38, 60, 56, 0, -42, -96, -44, -24, -18, 72, -48, 0, 7, -15, 72, -28, -54, 0, 72, 0, 80, 90, -60, 48, -62, 96, -24, 0, 84, -144, -68, -36, 96, -144, -72, 0, -74, 114, -20, -40, 96, -168

Offset

1,2

References

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

Andrew R. Feist, Fun With the Sigma-Function, unpub.

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Andrew Howroyd)

G. P. Brown, Some comments on inverse arithmetic functions, Math. Gaz. 89 (516) (2005) 403-408.

Formula

a(p) = -p-1, a(p^2) = p, a(p^k) = 0 for k > 2.

Dirichlet g.f.: 1/(zeta(s)*zeta(s-1)). - Benedict W. J. Irwin, Jul 10 2018

G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} sigma(k)*A(x^k). - Ilya Gutkovskiy, May 11 2019

From Peter Bala, Jan 17 2024: (Start)

a(n) = Sum_{d divides n} d*mu(d)*mu(n/d). See Brown, p. 408.

a(n) = - Sum_{d divides n, d < n} a(d)*sigma_1(n/d).

a(n) = Sum_{d divides n} d*a(d)*J_2(n/d), where the Jordan totient function J_2(n) = A007434(n).

a(n) = Sum_{d divides n} d*A007427(d)*phi(n/d), where A007427 is the Dirichlet inverse of the tau function.

More generally, a(n) = Sum_{d divides n} d*sigma_[r]^(-1)(d)*J_(r+1)(n/d), where sigma_[r]^(-1) denotes the Dirichlet inverse of the function sigma_[r] = Sum_{d divides n} d^r.

a(n) = Sum_{k = 1..n} gcd(k, n)*A007427(gcd(k, n)).

a(n) = Sum_{1 <= j, k <= n} gcd(j, k, n)*a(gcd(j, k, n)). (End)

Sum_{k=1..n} abs(a(k)) ~ 45*n^2/Pi^4. - Vaclav Kotesovec, May 30 2024

Example

a(36) = a(2^2*3^2) = 2*3 = 6.

Maple

t := 1; a := proc(n, t) local t1, d; t1 := 0; for d from 1 to n do if n mod d = 0 then t1 := t1+d^t*mobius(d)*mobius(n/d); fi; od; t1; end;

Mathematica

a[n_] := (k = 0; Do[If[Mod[n, d] == 0, k = k + d*MoebiusMu[d]*MoebiusMu[n/d]], {d, 1, n}]; k); Table[a[n], {n, 1, 78}](* Jean-François Alcover, Oct 13 2011, after Maple *)
f[p_, e_] := Which[e == 1, -p-1, e == 2, p, e >= 3, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 16 2020 *)

Prog

(PARI) a(n)=if(n<1, 0, direuler(p=2, n, (1-X)*(1-p*X))[n]) /* Ralf Stephan */
(PARI) seq(n)={dirdiv(vector(n, n, n==1), vector(n, n, sigma(n)))} \\ Andrew Howroyd, Aug 05 2018

Crossrefs

Cf. A000203, A053822, A053825, A053826, A178448.

Sequence in context: A269868 A344968 A324340 * A205769 A166108 A255768

Adjacent sequences: A046689 A046690 A046691 * A046693 A046694 A046695

Keyword

easy,mult,sign,nice

Author

Andrew R. Feist (arf22540(AT)cmsu2.cmsu.edu)

Extensions

Corrected by T. D. Noe, Nov 13 2006

Status

approved