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

Author

Amiram Eldar, Dec 30 2024

Keywords

• nonn
• easy

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.

Crossrefs

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

Programs

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]

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)));

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