A379832 The second Jordan totient function applied to the exponentially odd numbers.

1, 3, 8, 24, 24, 48, 48, 72, 120, 168, 144, 192, 288, 360, 384, 360, 528, 384, 504, 648, 840, 576, 960, 768, 960, 864, 1152, 1368, 1080, 1344, 1152, 1680, 1152, 1848, 1584, 2208, 2304, 2808, 1944, 2880, 2304, 2880, 2520, 3480, 3720, 2880, 4032, 2880, 4488, 4224

Offset: 1

Author

Amiram Eldar, Jan 03 2025

Keywords

• nonn
• easy

Links

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

Crossrefs

Cf. A007434, A065463, A185197, A268335, A374456 (analogous with J_1 = phi), A379715, A379716, A379717, A379718, A379833.

Programs

Mathematica

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

PARI

j2(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^2 - 1) * f[i,1]^(2*f[i,2] - 2));}
isexpodd(n) = {my(f = factor(n)); for(i=1, #f~, if(!(f[i, 2] % 2), return (0))); 1;}
list(lim) = apply(j2, select(isexpodd, vector(lim, i, i)));

Formula

a(n) = A007434(A268335(n)).

Sum_{k=1..n} a(k) ~ c * n^3, where c = 2/(Pi^2 * Product_{p prime} (1 - 1/(p*(p+1)))^3) = A185197 / A065463^3 = 0.57968779180803379088... .

Sum_{n>=1} 1/a(n) = (Pi^6/540) * Product_{p prime} (1 - 1/p^4 + 1/p^6) = 1.67479534964539923068...

In general, Sum_{m exponentially odd} 1/J_k(m) = zeta(k) * zeta(2*k) * Product_{p prime} (1 - 1/p^(2*k) + 1/p^(3*k)), for k >= 2, where J_k is the k-th Jordan totient function.