%I #32 Jan 27 2024 10:30:53

%S 1,-17,-82,16,-626,1394,-2402,0,81,10642,-14642,-1312,-28562,40834,

%T 51332,0,-83522,-1377,-130322,-10016,196964,248914,-279842,0,625,

%U 485554,0,-38432,-707282,-872644,-923522,0,1200644,1419874,1503652,1296,-1874162

%N Dirichlet inverse of sigma_4 function (A001159).

%C sigma_4(n) is the sum of the 4th powers of the divisors of n (A001159).

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

%H G. C. Greubel, <a href="/A053826/b053826.txt">Table of n, a(n) for n = 1..10000</a>

%F Dirichlet g.f.: 1/(zeta(x)*zeta(x-4)).

%F Multiplicative with a(p^1) = -1 - p^4, a(p^2) = p^4, a(p^e) = 0 for e>=3. - _Mitch Harris_, Jun 27 2005

%F a(n) = Sum_{d|n} mu(n/d)*mu(d)*d^4. - _Ilya Gutkovskiy_, Nov 06 2018

%F From _Peter Bala_, Jan 17 2024: (Start)

%F a(n) = Sum_{d divides n} d * A053825(d) * phi(n/d), where the totient function phi(n) = A000010(n).

%F a(n) = Sum_{d divides n} d^2 * (sigma_2(d))^(-1) * J_2(n/d),

%F a(n) = Sum_{d divides n} d^3 * (sigma_1(d))^(-1) * J_3(n/d), and for k >= 0,

%F a(n) = Sum_{d divides n} d^4 * (sigma_k(d))^(-1) * J_(k+4)(n/d), 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)

%t Table[DivisorSum[n, MoebiusMu[n/#]*MoebiusMu[#]*#^4 &], {n, 1, 50}] (* _G. C. Greubel_, Nov 07 2018 *)

%t f[p_, e_] := If[e == 1, -p^4 - 1, If[e == 2, p^4, 0]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 16 2020 *)

%o (PARI) a(n) = sumdiv(n, d, moebius(n/d)*moebius(d)*d^4); \\ _Michel Marcus_, Nov 06 2018

%o (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 - X)*(1 - p^4*X))[n], ", ")) \\ _Vaclav Kotesovec_, Sep 16 2020

%Y Dirichlet inverse of sigma_k(n): A007427 (k = 0), A046692 (k = 1), A053822(k = 2), A053825 (k = 3), A178448 (k = 5).

%Y Cf. A001159, A046099 (where a(n) = 0).

%K sign,mult

%O 1,2

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