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A194532 Jordan function ratio J_6(n)/J_2(n).

%I A194532 #28 Nov 05 2022 08:18:41
%S A194532 1,21,91,336,651,1911,2451,5376,7371,13671,14763,30576,28731,51471,
%T A194532 59241,86016,83811,154791,130683,218736,223041,310023,280371,489216,
%U A194532 406875,603351,597051,823536,708123,1244061,924483,1376256,1343433,1760031,1595601,2476656,1875531,2744343
%N A194532 Jordan function ratio J_6(n)/J_2(n).
%C A194532 Dirichlet convolution of A000583 with the multiplicative function which starts 1, 5, 10, 0, 26, 50, 50, 0, 0, 130, 122, 0, 170, 250, 260, 0, 290,..
%H A194532 Robert Israel, <a href="/A194532/b194532.txt">Table of n, a(n) for n = 1..10000</a>
%F A194532 a(n) = A069091(n)/A007434(n).
%F A194532 Multiplicative with a(p^e) = p^(4*(e-1))*(p^2+p+1)*(p^2-p+1), e>0.
%F A194532 Dirichlet g.f.: zeta(s-4)*product_{primes p} (1+p^(2-s)+p^(-s)).
%F A194532 Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = Product_{primes p} (1 + 1/p^3 + 1/p^5) = 1.2196771388395597011492820972459808778277319864216893177353903924... - _Vaclav Kotesovec_, Dec 18 2019
%F A194532 Sum_{n>=1} 1/a(n) = (Pi^8/14175) * Product_{p prime} (1 + 1/p^2 + 1/p^4 - 1/p^6 - 1/p^8) = 1.06469274411... . - _Amiram Eldar_, Nov 05 2022
%p A194532 f:= proc(n) local t;
%p A194532      mul(t[1]^(4*(t[2]-1))*((t[1]^2+1)^2-t[1]^2),t=ifactors(n)[2])
%p A194532 end proc:
%p A194532 map(f, [$1..100]); # _Robert Israel_, Jun 14 2016
%t A194532 JordanTotient[n_, k_: 1] := DivisorSum[n, #^k MoebiusMu[n/#] &] /; (n > 0) && IntegerQ@ n; Table[JordanTotient[n, 6]/JordanTotient[n, 2], {n, 12}] (* _Michael De Vlieger_, Jun 14 2016, after _Enrique Pérez Herrero_ at A065959 *)
%t A194532 f[p_, e_] := p^(4*(e-1))*(p^2+p+1)*(p^2-p+1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Oct 12 2020 *)
%o A194532 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^(4*(f[i,2]-1))*(f[i,1]^2+f[i,1]+1)*(f[i,1]^2-f[i,1]+1));} \\ _Amiram Eldar_, Nov 05 2022
%Y A194532 Cf. A007434, A065959, A069091.
%K A194532 nonn,mult,easy
%O A194532 1,2
%A A194532 _R. J. Mathar_, Aug 28 2011