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A379715 The second Jordan totient function applied to the squarefree numbers.

%I A379715 #14 Jan 04 2025 02:48:29
%S A379715 1,3,8,24,24,48,72,120,168,144,192,288,360,384,360,528,504,840,576,
%T A379715 960,960,864,1152,1368,1080,1344,1680,1152,1848,1584,2208,2304,2808,
%U A379715 2880,2880,2520,3480,3720,2880,4032,2880,4488,4224,3456,5040,5328,4104,5760,4032
%N A379715 The second Jordan totient function applied to the squarefree numbers.
%H A379715 Amiram Eldar, <a href="/A379715/b379715.txt">Table of n, a(n) for n = 1..10000</a>
%H A379715 Mohammadreza Esfandiari, <a href="https://doi.org/10.1007/s41980-020-00356-y">On the Means of Jordan's Totient Function</a>, Bull. Iran. Math. Soc., Vol. 46 (2020), pp. 1753-1765.
%H A379715 R. Sitaramachandrarao, <a href="https://doi.org/10.1216/RMJ-1985-15-2-579">On an error term of Landau - II</a>, Rocky Mountain J. Math., Vol. 15, No. 2 (1985), pp. 579-588. See p. 581.
%F A379715 a(n) = A007434(A005117(n)).
%F A379715 Sum_{n>=1} 1/a(n) = zeta(2) (A013661) (Sitaramachandrarao, 1985).
%F A379715 In general, Sum_{m squarefree} 1/J_k(m) = zeta(k), for k >= 2, where J_k is the k-th Jordan totient function.
%F A379715 Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = zeta(2)^3 * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) =  A013661^3 * A330523 = 2.38520727393117206135... . - _Amiram Eldar_, Jan 03 2025
%t A379715 f[p_, e_] := (p^2-1) * p^(2*e-2); j2[1] = 1; j2[n_] := Times @@ f @@@ FactorInteger[n]; j2 /@ Select[Range[100], SquareFreeQ]
%o A379715 (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 A379715 list(lim) = apply(j2, select(issquarefree, vector(lim, i, i)));
%Y A379715 Cf. A005117, A007434, A013661, A049200 (analogous with J_1 = phi), A330523, A379716, A379717, A379718.
%K A379715 nonn,easy
%O A379715 1,2
%A A379715 _Amiram Eldar_, Dec 30 2024