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A379832 The second Jordan totient function applied to the exponentially odd numbers.

%I A379832 #16 Jan 17 2025 03:38:47
%S A379832 1,3,8,24,24,48,48,72,120,168,144,192,288,360,384,360,528,384,504,648,
%T A379832 840,576,960,768,960,864,1152,1368,1080,1344,1152,1680,1152,1848,1584,
%U A379832 2208,2304,2808,1944,2880,2304,2880,2520,3480,3720,2880,4032,2880,4488,4224
%N A379832 The second Jordan totient function applied to the exponentially odd numbers.
%H A379832 Amiram Eldar, <a href="/A379832/b379832.txt">Table of n, a(n) for n = 1..10000</a>
%F A379832 a(n) = A007434(A268335(n)).
%F A379832 Sum_{k=1..n} a(k) ~ c * n^3, where c = 2/(Pi^2 * Product_{p prime} (1 - 1/(p*(p+1)))^3) = A185197 / A065463^3 = 0.57968779180803379088... .
%F A379832 Sum_{n>=1} 1/a(n) = (Pi^6/540) * Product_{p prime} (1 - 1/p^4 + 1/p^6) = 1.67479534964539923068...
%F A379832 In general, Sum_{m exponentially odd} 1/J_k(m) = zeta(k) * zeta(2*k) * Product_{p prime} (1 - 1/p^(2*k) + 1/p^(3*k)), for k >= 2, where J_k is the k-th Jordan totient function.
%t A379832 f[p_, e_] := (p^2-1) * p^(2*e-2); j2[1] = 1; j2[n_] := Times @@ f @@@ FactorInteger[n]; expoddQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], OddQ]; j2 /@ Select[Range[100], expoddQ]
%o A379832 (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 A379832 isexpodd(n) = {my(f = factor(n)); for(i=1, #f~, if(!(f[i, 2] % 2), return (0))); 1;}
%o A379832 list(lim) = apply(j2, select(isexpodd, vector(lim, i, i)));
%Y A379832 Cf. A007434, A065463, A185197, A268335, A374456 (analogous with J_1 = phi), A379715, A379716, A379717, A379718, A379833.
%K A379832 nonn,easy
%O A379832 1,2
%A A379832 _Amiram Eldar_, Jan 03 2025