A379718 - OEIS

%I #10 Jan 01 2025 14:42:01

%S 1,48,192,648,768,3072,5832,15000,12288,31104,52488,49152,115248,

%T 124416,196608,375000,279936,472392,497664,720000,786432,1119744,

%U 1756920,1990656,2519424,2880000,3145728,4251528,4798248,5647152,4478976,5531904,9375000,9720000,7962624

%N The second Jordan totient function applied to the cubefull numbers: a(n) = A007434(A036966(n)).

%H Amiram Eldar, <a href="/A379718/b379718.txt">Table of n, a(n) for n = 1..10000</a>

%F Sum_{n>=1} 1/a(n) = zeta(2)^2 * Product_{p prime} (1 - 2/p^2 + 1/p^4 + 1/p^6) = 1.02964361441212748276... .

%F In general, Sum_{m cubefull} 1/J_k(m) = zeta(k)^2 * Product_{p prime} (1 - 2/p^k + 1/p^(2*k) + 1/p^(3*k)), for k >= 2, where J_k is the k-th Jordan totient function.

%F In general, Sum_{m k-full} 1/J_2(m) = zeta(2)^2 * Product_{p prime} (1 - 2/p^2 + 1/p^4 + 1/p^(2*k)), for k >= 2.

%t f[p_, e_] := (p^2-1) * p^(2*e-2); j2[1] = 1; j2[n_] := Times @@ f @@@ FactorInteger[n]; Join[{1}, j2 /@ Select[Range[20000], AllTrue[Last /@ FactorInteger[#], #1 > 2 &] &]]

%o (PARI) j2(f) = prod(i = 1, #f~, (f[i,1]^2 - 1) * f[i,1]^(2*f[i,2] - 2));

%o list(lim) = {my(f); print1(1, ", "); for(k = 2, lim, f = factor(k); if(vecmin(f[, 2]) > 2, print1(j2(f), ", "))); }

%Y Cf. A007434, A013661, A036966, A371412 (analogous with J_1 = phi), A379715, A379716, A379717.

%K nonn,easy,new

%O 1,2

%A _Amiram Eldar_, Dec 31 2024