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

1,2

Links

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

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

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]

Prog

(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)));

Crossrefs

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

Sequence in context: A158022 A209934 A007434 * A128303 A123906 A395181

Adjacent sequences: A379714 A379715 A379716 * A379718 A379719 A379720

Keyword

nonn,easy

Author

Amiram Eldar, Dec 31 2024

Status

approved