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A046692 Dirichlet inverse of sigma function (A000203).

%I A046692 #44 May 30 2024 17:56:20
%S A046692 1,-3,-4,2,-6,12,-8,0,3,18,-12,-8,-14,24,24,0,-18,-9,-20,-12,32,36,
%T A046692 -24,0,5,42,0,-16,-30,-72,-32,0,48,54,48,6,-38,60,56,0,-42,-96,-44,
%U A046692 -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
%N A046692 Dirichlet inverse of sigma function (A000203).
%D A046692 Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 39.
%D A046692 Andrew R. Feist, Fun With the Sigma-Function, unpub.
%H A046692 Seiichi Manyama, <a href="/A046692/b046692.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Andrew Howroyd)
%H A046692 G. P. Brown, <a href="http://www.jstor.org/stable/3621931">Some comments on inverse arithmetic functions</a>, Math. Gaz. 89 (516) (2005) 403-408.
%F A046692 a(p) = -p-1, a(p^2) = p, a(p^k) = 0 for k > 2.
%F A046692 Dirichlet g.f.: 1/(zeta(s)*zeta(s-1)). - _Benedict W. J. Irwin_, Jul 10 2018
%F A046692 G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} sigma(k)*A(x^k). - _Ilya Gutkovskiy_, May 11 2019
%F A046692 From _Peter Bala_, Jan 17 2024: (Start)
%F A046692 a(n) = Sum_{d divides n} d*mu(d)*mu(n/d). See Brown, p. 408.
%F A046692 a(n) = - Sum_{d divides n, d < n} a(d)*sigma_1(n/d).
%F A046692 a(n) = Sum_{d divides n} d*a(d)*J_2(n/d), where the Jordan totient function J_2(n) = A007434(n).
%F A046692 a(n) = Sum_{d divides n} d*A007427(d)*phi(n/d), where A007427 is the Dirichlet inverse of the tau function.
%F A046692 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.
%F A046692 a(n) = Sum_{k = 1..n} gcd(k, n)*A007427(gcd(k, n)).
%F A046692 a(n) = Sum_{1 <= j, k <= n} gcd(j, k, n)*a(gcd(j, k, n)). (End)
%F A046692 Sum_{k=1..n} abs(a(k)) ~ 45*n^2/Pi^4. - _Vaclav Kotesovec_, May 30 2024
%e A046692 a(36) = a(2^2*3^2) = 2*3 = 6.
%p A046692 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;
%t A046692 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 *)
%t A046692 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 *)
%o A046692 (PARI) a(n)=if(n<1,0,direuler(p=2,n,(1-X)*(1-p*X))[n]) /* _Ralf Stephan_ */
%o A046692 (PARI) seq(n)={dirdiv(vector(n, n, n==1), vector(n, n, sigma(n)))} \\ _Andrew Howroyd_, Aug 05 2018
%Y A046692 Cf. A000203, A053822, A053825, A053826, A178448.
%K A046692 easy,mult,sign,nice
%O A046692 1,2
%A A046692 Andrew R. Feist (arf22540(AT)cmsu2.cmsu.edu)
%E A046692 Corrected by _T. D. Noe_, Nov 13 2006