This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A379716 #10 Jan 01 2025 14:41:51
%S A379716 1,12,48,72,192,600,648,768,864,2352,3072,3456,5832,7200,7776,14520,
%T A379716 15000,12288,13824,28392,28224,28800,31104,43200,52488,49152,55296,
%U A379716 83232,69984,115248,129960,112896,115200,124416,169344,174240,180000,196608,279312,221184,375000
%N A379716 The second Jordan totient function applied to the powerful numbers: a(n) = A007434(A001694(n)).
%H A379716 Amiram Eldar, <a href="/A379716/b379716.txt">Table of n, a(n) for n = 1..10000</a>
%F A379716 Sum_{n>=1} 1/a(n) = zeta(2)^2 * Product_{p prime} (1 - 2/p^2 + 2/p^4) = 1.13107206648894940601... .
%F A379716 In general, Sum_{m powerful} 1/J_k(m) = zeta(k)^2 * Product_{p prime} (1 - 2/p^k + 2/p^(2*k)), for k >= 2, where J_k is the k-th Jordan totient function.
%t A379716 f[p_, e_] := (p^2-1) * p^(2*e-2); j2[1] = 1; j2[n_] := Times @@ f @@@ FactorInteger[n]; seq[lim_] := j2 /@ Union[Flatten[Table[i^2*j^3, {j, 1, Surd[lim, 3]}, {i, 1, Sqrt[lim/j^3]}]]]; seq[1000]
%o A379716 (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 A379716 list(lim) = apply(j2, select(ispowerful, vector(lim, i, i)));
%Y A379716 Cf. A001694, A007434, A323333 (analogous with J_1 = phi), A379715, A379717, A379718.
%K A379716 nonn,easy
%O A379716 1,2
%A A379716 _Amiram Eldar_, Dec 30 2024