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

1,2

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.

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

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]

Prog

(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

Crossrefs

Cf. A072911.

Sequence in context: A254503 A186646 A309108 * A336965 A293303 A333569

Adjacent sequences: A308053 A308054 A308055 * A308057 A308058 A308059

Keyword

nonn,mult

Author

Amiram Eldar, May 10 2019

Status

approved