A053822 - OEIS

%I #40 Jan 27 2024 10:30:40

%S 1,-5,-10,4,-26,50,-50,0,9,130,-122,-40,-170,250,260,0,-290,-45,-362,

%T -104,500,610,-530,0,25,850,0,-200,-842,-1300,-962,0,1220,1450,1300,

%U 36,-1370,1810,1700,0,-1682,-2500,-1850,-488,-234,2650,-2210,0,49,-125,2900,-680

%N Dirichlet inverse of sigma_2 function (A001157).

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

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

%H Robert Israel, <a href="/A053822/b053822.txt">Table of n, a(n) for n = 1..10000</a>

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

%F 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

%F a(n) = Sum_{d|n} mu(n/d)*mu(d)*d^2. - _Ilya Gutkovskiy_, Nov 06 2018

%F From _Peter Bala_, Jan 26 2024: (Start)

%F 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).

%F 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)

%p f1:= proc(p,e) if e = 1 then -1-p^2 elif e=2 then p^2 else 0 fi end proc:

%p f:= n -> mul(f1(t[1],t[2]),t=ifactors(n)[2]);

%p map(f, [$1..100]); # _Robert Israel_, Jan 29 2018

%t a[n_] := Sum[MoebiusMu[n/d] MoebiusMu[d] d^2, {d, Divisors[n]}];

%t Array[a, 100] (* _Jean-François Alcover_, Mar 05 2019, after _Ilya Gutkovskiy_ *)

%t 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 *)

%o (PARI) seq(n)={dirdiv(vector(n, n, n==1), vector(n, n, sigma(n, 2)))} \\ _Andrew Howroyd_, Aug 05 2018

%o (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 - X)*(1 - p^2*X))[n], ", ")) \\ _Vaclav Kotesovec_, Sep 16 2020

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

%Y Cf. A001157,.

%K sign,mult,look

%O 1,2

%A _N. J. A. Sloane_, Apr 08 2000