A379716 The second Jordan totient function applied to the powerful numbers: a(n) = A007434(A001694(n)).

1, 12, 48, 72, 192, 600, 648, 768, 864, 2352, 3072, 3456, 5832, 7200, 7776, 14520, 15000, 12288, 13824, 28392, 28224, 28800, 31104, 43200, 52488, 49152, 55296, 83232, 69984, 115248, 129960, 112896, 115200, 124416, 169344, 174240, 180000, 196608, 279312, 221184, 375000

Offset: 1

Author

Amiram Eldar, Dec 30 2024

Keywords

• nonn
• easy

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001694, A007434, A323333 (analogous with J_1 = phi), A379715, A379717, A379718.

Programs

Mathematica

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]

PARI

j2(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^2 - 1) * f[i,1]^(2*f[i,2] - 2));}
list(lim) = apply(j2, select(ispowerful, vector(lim, i, i)));

Formula

Sum_{n>=1} 1/a(n) = zeta(2)^2 * Product_{p prime} (1 - 2/p^2 + 2/p^4) = 1.13107206648894940601... .

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.