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%I #40 Jan 26 2024 09:18:12
%S 1,-15,-80,-15,-624,1200,-2400,-15,-80,9360,-14640,1200,-28560,36000,
%T 49920,-15,-83520,1200,-130320,9360,192000,219600,-279840,1200,-624,
%U 428400,-80,36000,-707280,-748800,-923520,-15,1171200,1252800,1497600,1200,-1874160
%N Jordan function J_{-4} multiplied by n^4.
%C For the Jordan function J_k see the Comtet and Apostol references.
%D T. M. Apostol, Introduction to Analytic Number Theory, Springer, 1986.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.
%H G. C. Greubel, <a href="/A189922/b189922.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..200 from Indranil Ghosh)
%F a(n) = J_{-4}(n)*n^4 = Product_{p prime | n} (1 - p^4), for n>=2, a(1)=1.
%F a(n) = Sum_{d|n} mu(d)*d^4 with the Moebius function mu = A008683.
%F Dirichlet g.f.: zeta(s)/zeta(s-4).
%F Sum identity: Sum_{d|n} a(n)*(n/d)^4 = 1 for all n>=1.
%F a(n) = a(rad(n)) with rad(n) = A007947(n), the squarefree kernel of n.
%F G.f.: Sum_{k>=1} mu(k)*k^4*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Jan 15 2017
%F a(n) = Sum_{d divides n} d * sigma_3(d)^(-1) * sigma_1(n/d), where sigma_3(n)^(-1) = A053825(n) denotes the Dirichlet inverse of sigma_3(n). - _Peter Bala_, Jan 26 2024
%e a(2) = a(4) = a(8) = ... = 1 - 2^4 = -15.
%e a(4) = mu(1)*1^4 + mu(2)*2^4 + mu(4)*4^4 = 1 - 16 + 0 = -15.
%e Sum identity for n=4: a(1)*(4/1)^4 + a(2)*(4/2)^4 + a(4)*(4/4)^4 = 256 - 15*16 - 15 = 1.
%p a:= n-> mul(1-i[1]^4, i=ifactors(n)[2]):
%p seq(a(n), n=1..48); # _Alois P. Heinz_, Jan 26 2024
%t a[n_] := Sum[ MoebiusMu[d]*d^4, {d, Divisors[n]}]; Table[a[n], {n, 1, 30}] (* _Jean-François Alcover_, Sep 03 2012 *)
%t f[p_, e_] := (1-p^4); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Dec 08 2020 *)
%o (PARI) for (n=1, 30, print1(sumdiv(n, d, moebius(d) * d^4),", ")); \\ _Indranil Ghosh_, Mar 11 2017
%Y Cf. A023900 (k=-1), A046970 (k=-2), A063453 (k=-3).
%K sign,easy,mult
%O 1,2
%A _Wolfdieter Lang_, Jun 16 2011
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