A379715 The second Jordan totient function applied to the squarefree numbers.

1, 3, 8, 24, 24, 48, 72, 120, 168, 144, 192, 288, 360, 384, 360, 528, 504, 840, 576, 960, 960, 864, 1152, 1368, 1080, 1344, 1680, 1152, 1848, 1584, 2208, 2304, 2808, 2880, 2880, 2520, 3480, 3720, 2880, 4032, 2880, 4488, 4224, 3456, 5040, 5328, 4104, 5760, 4032

Offset

1,2

Links

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

Mohammadreza Esfandiari, On the Means of Jordan's Totient Function, Bull. Iran. Math. Soc., Vol. 46 (2020), pp. 1753-1765.

R. Sitaramachandrarao, On an error term of Landau - II, Rocky Mountain J. Math., Vol. 15, No. 2 (1985), pp. 579-588. See p. 581.

Formula

a(n) = A007434(A005117(n)).

Sum_{n>=1} 1/a(n) = zeta(2) (A013661) (Sitaramachandrarao, 1985).

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

Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = zeta(2)^3 * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = A013661^3 * A330523 = 2.38520727393117206135... . - 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]; j2 /@ Select[Range[100], SquareFreeQ]

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)); }
list(lim) = apply(j2, select(issquarefree, vector(lim, i, i)));

Crossrefs

Cf. A005117, A007434, A013661, A049200 (analogous with J_1 = phi), A330523, A379716, A379717, A379718.

Sequence in context: A065083 A379832 A382661 * A280190 A037450 A081990

Adjacent sequences: A379712 A379713 A379714 * A379716 A379717 A379718

Keyword

nonn,easy

Author

Amiram Eldar, Dec 30 2024

Status

approved