A379717 The second Jordan totient function applied to the cubefree numbers.

1, 3, 8, 12, 24, 24, 48, 72, 72, 120, 96, 168, 144, 192, 288, 216, 360, 288, 384, 360, 528, 600, 504, 576, 840, 576, 960, 960, 864, 1152, 864, 1368, 1080, 1344, 1680, 1152, 1848, 1440, 1728, 1584, 2208, 2352, 1800, 2304, 2016, 2808, 2880, 2880, 2520, 3480, 2304

Offset: 1

Author

Amiram Eldar, Dec 31 2024

Keywords

• nonn
• easy

Links

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

Crossrefs

Cf. A002117, A004709, A007434, A013661, A358039 (analogous with J_1 = phi), A379715, A379716, A379718.

Programs

Mathematica

f[p_, e_] := (p^2-1) * p^(2*e-2); j2[1] = 1; j2[n_] := Times @@ f @@@ FactorInteger[n]; cubeFreeQ[n_] := Max[FactorInteger[n][[;;, 2]]] < 3; j2 /@ Select[Range[100], cubeFreeQ]

PARI

j2(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^2 - 1) * f[i,1]^(2*f[i,2] - 2));}
iscubefree(n) = if(n == 1, 1, vecmax(factor(n)[, 2]) < 3);
list(lim) = apply(j2, select(iscubefree, vector(lim, i, i)));

Formula

a(n) = A007434(A004709(n)).

Sum_{n>=1} 1/a(n) = zeta(2) * zeta(4) / zeta(8) = 35 / (2*Pi^2) = 1.77312071374091100026... .

In general, Sum_{m cubefree} 1/J_k(m) = zeta(k) * zeta(2*k) / zeta(4*k), for k >= 2, where J_k is the k-th Jordan totient function.

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

Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = zeta(3)^3 * Product_{p prime} (1 - 2/p^3 + 1/p^5) = 1.23061243656940899916... . - Amiram Eldar, Jan 03 2025