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%I #14 Feb 13 2024 02:20:17
%S 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,
%T 14,29,30,31,30,33,34,35,6,37,38,39,30,41,42,43,22,15,46,47,30,7,10,
%U 51,26,53,24,55,42,57,58,59,30,61,62,21,34,65,66,67,34,69
%N a(1) = 1, a(n) is the sum of the divisors d of n such that d and n are exponentially coprime.
%C The sequence of the number of those divisors is A072911.
%H Amiram Eldar, <a href="/A308056/b308056.txt">Table of n, a(n) for n = 1..10000</a>
%H László Tóth, <a href="http://ac.inf.elte.hu/Vol_024_2004/285.pdf">On certain arithmetic functions involving exponential divisors</a>, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 24 (2004), pp. 285-294; <a href="https://arxiv.org/abs/math/0610274">arXiv preprint</a>, arXiv:math/0610274v2 [math.NT], 2006-2009.
%F Multiplicative with a(p^e) = Sum_{i=1..e, gcd(i,e)=1} p^i.
%F 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
%t 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]
%o (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
%Y Cf. A072911.
%K nonn,mult
%O 1,2
%A _Amiram Eldar_, May 10 2019
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