A308056 a(1) = 1, a(n) is the sum of the divisors d of n such that d and n are exponentially coprime.

1, 2, 3, 2, 5, 6, 7, 6, 3, 10, 11, 6, 13, 14, 15, 10, 17, 6, 19, 10, 21, 22, 23, 18, 5, 26, 12, 14, 29, 30, 31, 30, 33, 34, 35, 6, 37, 38, 39, 30, 41, 42, 43, 22, 15, 46, 47, 30, 7, 10, 51, 26, 53, 24, 55, 42, 57, 58, 59, 30, 61, 62, 21, 34, 65, 66, 67, 34, 69

Offset: 1

Author

Amiram Eldar, May 10 2019

Keywords

• nonn
• mult

Comments

The sequence of the number of those divisors is A072911.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• László Tóth, On certain arithmetic functions involving exponential divisors, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 24 (2004), pp. 285-294; arXiv preprint, arXiv:math/0610274v2 [math.NT], 2006-2009.

Crossrefs

Cf. A072911.

Programs

Mathematica

fun[p_, e_] := Sum[If[GCD[i,e]==1, p^i, 0], {i,1,e}]; a[1] = 1; a[n_] := Times @@ (fun @@@ FactorInteger[n]); Array[a, 100]

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, sum(k = 1, f[i,2], (gcd(k, f[i, 2]) == 1) * f[i,1]^k));} \\ Amiram Eldar, Feb 13 2024

Formula

Multiplicative with a(p^e) = Sum_{i=1..e, gcd(i,e)=1} p^i.

Sum_{k=1..n} a(k) = c * n^2 / 2 + O(x^(3/2) * exp(A * log(n)^(3/5) * log(log(n))^(-1/5))), where A is a constant and c = Product_{p prime} (1 + Sum_{k>=2} (a(p^k) - p*a(p^(k-1)))/p^(2*k)) = 0.77693509739103041486... (Tóth, 2004). - Amiram Eldar, Feb 13 2024