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

%I A189923 #39 Dec 08 2020 03:49:21
%S A189923 1,-31,-242,-31,-3124,7502,-16806,-31,-242,96844,-161050,7502,-371292,
%T A189923 520986,756008,-31,-1419856,7502,-2476098,96844,4067052,4992550,
%U A189923 -6436342,7502,-3124,11510052,-242,520986,-20511148,-23436248
%N A189923 Jordan function J_{-5}(n) multiplied by n^5.
%C A189923 For the Jordan function J_k see the Comtet and Apostol references.
%D A189923 T. M. Apostol, Introduction to Analytic Number Theory, Springer, 1986.
%D A189923 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.
%H A189923 G. C. Greubel, <a href="/A189923/b189923.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..200 from Indranil Ghosh)
%F A189923 a(n) = J_{-5}(n)*n^5 = Product_{p prime |n} (1-p^5), for n>=2, a(1)=1.
%F A189923 a(n) = Sum_{d|n} mu(d)*d^5 with the Moebius function mu = A008683.
%F A189923 Dirichlet g.f.: zeta(s)/zeta(s-5).
%F A189923 Sum identity: Sum_{d|n} a(n)*(n/d)^5 = 1 for all n>=1.
%F A189923 a(n) = a(rad(n)) with rad(n) = A007947(n), the squarefree kernel of n.
%F A189923 G.f.: Sum_{k>=1} mu(k)*k^5*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Jan 15 2017
%e A189923 a(2) = a(4) = a(8) = ... = 1 - 2^5 = -31.
%e A189923 a(4) = mu(1)*1^5 + mu(2)*2^5 + mu(4)*4^5 = 1 - 32 + 0 = -31.
%e A189923 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.
%t A189923 a[n_] := Sum[ MoebiusMu[d]*d^5, {d, Divisors[n]}]; Table[a[n], {n, 1, 30}] (* _Jean-François Alcover_, Sep 03 2012 *)
%t A189923 f[p_, e_] := (1-p^5); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Dec 08 2020 *)
%o A189923 (PARI) for(n=1, 200, print1(sumdiv(n, d, moebius(d) * d^5),", ")) \\ _Indranil Ghosh_, Mar 11 2017
%o A189923 (PARI) a(n) = sumdiv(n, d, moebius(d) * d^5); \\ _Michel Marcus_, Jan 14 2018
%Y A189923 Cf. A023900, A046970, A063453, A189922, for k=-1..-4.
%K A189923 sign,easy,mult
%O A189923 1,2
%A A189923 _Wolfdieter Lang_, Jun 16 2011