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

1,2

Links

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

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.

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]

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)); }
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)));

Crossrefs

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

Sequence in context: A317362 A309114 A065083 * A379715 A280190 A037450

Adjacent sequences: A379829 A379830 A379831 * A379833 A379834 A379835

Keyword

nonn,easy

Author

Amiram Eldar, Jan 03 2025

Status

approved