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A379717 The second Jordan totient function applied to the cubefree numbers.

%I A379717 #17 Jan 04 2025 02:48:34
%S A379717 1,3,8,12,24,24,48,72,72,120,96,168,144,192,288,216,360,288,384,360,
%T A379717 528,600,504,576,840,576,960,960,864,1152,864,1368,1080,1344,1680,
%U A379717 1152,1848,1440,1728,1584,2208,2352,1800,2304,2016,2808,2880,2880,2520,3480,2304
%N A379717 The second Jordan totient function applied to the cubefree numbers.
%H A379717 Amiram Eldar, <a href="/A379717/b379717.txt">Table of n, a(n) for n = 1..10000</a>
%F A379717 a(n) = A007434(A004709(n)).
%F A379717 Sum_{n>=1} 1/a(n) = zeta(2) * zeta(4) / zeta(8) = 35 / (2*Pi^2) = 1.77312071374091100026... .
%F A379717 In general, Sum_{m cubefree} 1/J_k(m) = zeta(k) * zeta(2*k) / zeta(4*k), for k >= 2, where J_k is the k-th Jordan totient function.
%F A379717 In general, Sum_{m k-free} 1/J_2(m) = zeta(2)^2 * Product_{p prime} (1 - 1/p^2 + 1/p^4 - 1/p^(2*k)), for k >= 2.
%F A379717 Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = zeta(3)^3 * Product_{p prime} (1 - 2/p^3 + 1/p^5) = 1.23061243656940899916... . - _Amiram Eldar_, Jan 03 2025
%t A379717 f[p_, e_] := (p^2-1) * p^(2*e-2); j2[1] = 1; j2[n_] := Times @@ f @@@ FactorInteger[n]; cubeFreeQ[n_] := Max[FactorInteger[n][[;;, 2]]] < 3; j2 /@ Select[Range[100], cubeFreeQ]
%o A379717 (PARI) j2(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^2 - 1) * f[i,1]^(2*f[i,2] - 2));}
%o A379717 iscubefree(n) = if(n == 1, 1, vecmax(factor(n)[, 2]) < 3);
%o A379717 list(lim) = apply(j2, select(iscubefree, vector(lim, i, i)));
%Y A379717 Cf. A002117, A004709, A007434, A013661, A358039 (analogous with J_1 = phi), A379715, A379716, A379718.
%K A379717 nonn,easy
%O A379717 1,2
%A A379717 _Amiram Eldar_, Dec 31 2024