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A381713 a(n) = J_9(n)/J_3(n), where J_k is the k-th Jordan totient function.

%I A381713 #23 Mar 06 2025 01:49:32
%S A381713 1,73,757,4672,15751,55261,117993,299008,551853,1149823,1772893,
%T A381713 3536704,4829007,8613489,11923507,19136512,24142483,40285269,47052741,
%U A381713 73588672,89320701,129421189,148048057,226349056,246109375,352517511,402300837,551263296
%N A381713 a(n) = J_9(n)/J_3(n), where J_k is the k-th Jordan totient function.
%H A381713 Seiichi Manyama, <a href="/A381713/b381713.txt">Table of n, a(n) for n = 1..10000</a>
%F A381713 a(n) = A069094(n)/A059376(n).
%F A381713 a(n) = n^6 * Product_{distinct primes p dividing n} (1 + 1/p^3 + 1/p^6).
%F A381713 From _Amiram Eldar_, Mar 05 2025: (Start)
%F A381713 Dirichlet g.f.: zeta(s-6) * Product_{p prime} (1 + 1/p^(s-3) + 1/p^s).
%F A381713 Sum_{k=1..n} a(k) ~ c * n^7 / 7, where c = Product_{p prime} (1 + 1/p^4 + 1/p^7) = 1.08635980686198102055... .
%F A381713 Sum_{n>=1} 1/a(n) = zeta(6)*zeta(9) * Product_{p prime} (1 - 2/p^9 + 1/p^15) = 1.01533121878447451064... . (End)
%t A381713 f[p_, e_] := p^(6*e) * (1 + 1/p^3 + 1/p^6); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 28] (* _Amiram Eldar_, Mar 05 2025 *)
%o A381713 (PARI) J(n, k) = sumdiv(n, d, d^k*moebius(n/d));
%o A381713 a(n) = J(n, 9)/J(n, 3);
%o A381713 (PARI) a(n) = {my(p = factor(n)[, 1]); n^6 * prod(i = 1, #p, 1 + 1/p[i]^3 + 1/p[i]^6);} \\ _Amiram Eldar_, Mar 05 2025
%Y A381713 Cf. A065958, A160889, A194532.
%Y A381713 Cf. A059376, A069094.
%Y A381713 Cf. A013664, A013667.
%K A381713 nonn,mult
%O A381713 1,2
%A A381713 _Seiichi Manyama_, Mar 05 2025