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A069094 Jordan function J_9(n).

%I A069094 #32 Oct 12 2020 02:23:44
%S A069094 1,511,19682,261632,1953124,10057502,40353606,133955584,387400806,
%T A069094 998046364,2357947690,5149441024,10604499372,20620692666,38441386568,
%U A069094 68585259008,118587876496,197961811866,322687697778,510999738368
%N A069094 Jordan function J_9(n).
%D A069094 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.
%H A069094 G. C. Greubel, <a href="/A069094/b069094.txt">Table of n, a(n) for n = 1..5000</a>
%H A069094 Wikipedia, <a href="https://en.wikipedia.org/wiki/Jordan%27s_totient_function">Jordan's totient function</a>.
%F A069094 a(n) = Sum_{d|n} d^9*mu(n/d).
%F A069094 Multiplicative with a(p^e) = p^(9e)-p^(9(e-1)).
%F A069094 Dirichlet generating function: zeta(s-9)/zeta(s). - _Ralf Stephan_, Jul 04 2013
%F A069094 a(n) = n^9*Product_{distinct primes p dividing n} (1-1/p^9). - _Tom Edgar_, Jan 09 2015
%F A069094 Sum_{k=1..n} a(k) ~ 18711*n^10 / (2*Pi^10). - _Vaclav Kotesovec_, Feb 07 2019
%F A069094 From _Amiram Eldar_, Oct 12 2020: (Start)
%F A069094 lim_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k^9 = 1/zeta(10).
%F A069094 Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p^9/(p^9-1)^2) = 1.0020122252...  (End)
%t A069094 JordanJ[n_, k_] := DivisorSum[n, #^k*MoebiusMu[n/#] &]; f[n_] := JordanJ[n, 9]; Array[f, 22]
%t A069094 f[p_, e_] := p^(9*e) - p^(9*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Oct 12 2020 *)
%o A069094 (PARI) for(n=1,100,print1(sumdiv(n,d,d^9*moebius(n/d)),","))
%Y A069094 Cf. A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A059376 (J_3), A059377 (J_4), A059378 (J_5).
%Y A069094 Cf. A013668.
%K A069094 easy,nonn,mult
%O A069094 1,2
%A A069094 _Benoit Cloitre_, Apr 05 2002