A189923 Jordan function J_{-5}(n) multiplied by n^5.

1, -31, -242, -31, -3124, 7502, -16806, -31, -242, 96844, -161050, 7502, -371292, 520986, 756008, -31, -1419856, 7502, -2476098, 96844, 4067052, 4992550, -6436342, 7502, -3124, 11510052, -242, 520986, -20511148, -23436248

Offset

1,2

Comments

For the Jordan function J_k see the Comtet and Apostol references.

References

T. M. Apostol, Introduction to Analytic Number Theory, Springer, 1986.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000 (terms 1..200 from Indranil Ghosh)

Formula

a(n) = J_{-5}(n)*n^5 = Product_{p prime |n} (1-p^5), for n>=2, a(1)=1.

a(n) = Sum_{d|n} mu(d)*d^5 with the Moebius function mu = A008683.

Dirichlet g.f.: zeta(s)/zeta(s-5).

Sum identity: Sum_{d|n} a(n)*(n/d)^5 = 1 for all n>=1.

a(n) = a(rad(n)) with rad(n) = A007947(n), the squarefree kernel of n.

G.f.: Sum_{k>=1} mu(k)*k^5*x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 15 2017

Example

a(2) = a(4) = a(8) = ... = 1 - 2^5 = -31.
a(4) = mu(1)*1^5 + mu(2)*2^5 + mu(4)*4^5 = 1 - 32 + 0 = -31.
Sum identity for n=4: a(1)*(4/1)^5 + a(2)*(4/2)^5 + a(4)*(4/4)^5 = 1024 - 31*32 - 31 = 1.

Mathematica

a[n_] := Sum[ MoebiusMu[d]*d^5, {d, Divisors[n]}]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Sep 03 2012 *)
f[p_, e_] := (1-p^5); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 08 2020 *)

Prog

(PARI) for(n=1, 200, print1(sumdiv(n, d, moebius(d) * d^5), ", ")) \\ Indranil Ghosh, Mar 11 2017
(PARI) a(n) = sumdiv(n, d, moebius(d) * d^5); \\ Michel Marcus, Jan 14 2018

Crossrefs

Cf. A023900, A046970, A063453, A189922, for k=-1..-4.

Sequence in context: A173832 A272162 A338893 * A059378 A024003 A258807

Adjacent sequences: A189920 A189921 A189922 * A189924 A189925 A189926

Keyword

sign,easy,mult

Author

Wolfdieter Lang, Jun 16 2011

Status

approved