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A379833 The second Jordan totient function applied to the squares.

%I A379833 #13 Jan 17 2025 03:38:42
%S A379833 1,12,72,192,600,864,2352,3072,5832,7200,14520,13824,28392,28224,
%T A379833 43200,49152,83232,69984,129960,115200,169344,174240,279312,221184,
%U A379833 375000,340704,472392,451584,706440,518400,922560,786432,1045440,998784,1411200,1119744,1872792
%N A379833 The second Jordan totient function applied to the squares.
%H A379833 Amiram Eldar, <a href="/A379833/b379833.txt">Table of n, a(n) for n = 1..10000</a>
%F A379833 a(n) = J_2(n^2) = A007434(A000290(n)).
%F A379833 Multiplicative with a(p^e) = (p^2-1) * p^(4*e-2).
%F A379833 Dirichlet g.f.: zeta(s-4)/zeta(s-2).
%F A379833 In general, Dirichlet g.f. of J_k(n^m): zeta(s-m*k)/zeta(s-m*k+k), where J_k is the k-th Jordan totient function.
%F A379833 Sum_{i=1..n} a(i) ~ n^5 / (5*zeta(3)).
%F A379833 In general, Sum_{i=1..n} J_k(i^m) ~ n^(k*m+1) / ((k*m+1)*zeta(k+1)) for k,m >= 1.
%F A379833 Sum_{n>=1} 1/a(n) = (Pi^6/540) * Product_{p prime} (1 - 1/p^2 + 1/p^6) = 1.10666099915727116962...
%F A379833 In general, Sum_{n>=1} 1/J_k(n^m) = zeta(k) * zeta(k*m) * Product_{p prime} (1 - 1/p^k + 1/p^(k*m+k)), for k,m >= 2, and zeta(2) * zeta(m) * Product_{p prime} (1 - 1/p^2 + 1/p^(m+1) + 1/p^(m+2)) for k = 1 and m >= 2.
%p A379833 a:= n-> mul((i[1]^2-1)*i[1]^(4*i[2]-2), i=ifactors(n)[2]):
%p A379833 seq(a(n), n=1..37);  # _Alois P. Heinz_, Jan 03 2025
%t A379833 f[p_, e_] := (p^2 - 1) * p^(4*e - 2); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50]
%o A379833 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^2 - 1) * f[i,1]^(4*f[i,2] - 2));}
%Y A379833 Cf. A000290, A002117, A002618 (analogous with J_1 = phi), A007434, A379715, A379716, A379717, A379718, A379832.
%K A379833 nonn,easy,mult
%O A379833 1,2
%A A379833 _Amiram Eldar_, Jan 03 2025