A053826 Dirichlet inverse of sigma_4 function (A001159).

1, -17, -82, 16, -626, 1394, -2402, 0, 81, 10642, -14642, -1312, -28562, 40834, 51332, 0, -83522, -1377, -130322, -10016, 196964, 248914, -279842, 0, 625, 485554, 0, -38432, -707282, -872644, -923522, 0, 1200644, 1419874, 1503652, 1296, -1874162

Offset: 1

Author

N. J. A. Sloane, Apr 08 2000

Keywords

• sign
• mult

Comments

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

References

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

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000

Crossrefs

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

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

Programs

Mathematica

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

PARI

a(n) = sumdiv(n, d, moebius(n/d)*moebius(d)*d^4); \\ Michel Marcus, Nov 06 2018

PARI

for(n=1, 100, print1(direuler(p=2, n, (1 - X)*(1 - p^4*X))[n], ", ")) \\ Vaclav Kotesovec, Sep 16 2020

Formula

Dirichlet g.f.: 1/(zeta(s)*zeta(s-4)).

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

a(n) = Sum_{d|n} mu(n/d)*mu(d)*d^4. - Ilya Gutkovskiy, Nov 06 2018

From Peter Bala, Jan 17 2024: (Start)

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

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

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

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)