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%I #17 Oct 03 2023 13:14:17
%S 1,2,3,2,5,6,7,6,6,10,11,6,13,14,15,12,17,12,19,10,21,22,23,18,20,26,
%T 24,14,29,30,31,30,33,34,35,12,37,38,39,30,41,42,43,22,30,46,47,36,42,
%U 40,51,26,53,48,55,42,57,58,59,30,61,62,42,54,65,66,67
%N Exponential convolution of the exponential Mobius function and the natural numbers.
%C Exponential convolution of A166234 and A000027.
%C Similar to the definition of A000010 as the Dirichlet convolution of A008683 and A000027.
%H Amiram Eldar, <a href="/A293303/b293303.txt">Table of n, a(n) for n = 1..10000</a>
%H Andrew V. Lelechnko, <a href="http://arxiv.org/abs/1405.7597">Exponenital and infinitary divisors</a>, arXiv:1405.7597 [math.NT], 2014, sequence f^E(n).
%F Multiplicative with a(p^e) = Sum_{d|e} A008683(e/d)*p^d.
%F Sum_{k=1..n} a(k) ~ c * n^2, where c = 0.43802998037163511363... = (1/2) * Product_{p prime} (1-1/p)*Sum_{k>=1} (Sum_{d|e} mu(k/d)*p^k/p^(2*k)). - _Amiram Eldar_, Oct 03 2023
%p A293303 := proc(n)
%p local a,pe,i,p,e,f,d ;
%p a := 1 ;
%p for pe in ifactors(n)[2] do
%p p := pe[1] ;
%p e := pe[2] ;
%p f := 0 ;
%p for d in numtheory[divisors](e) do
%p f := f+numtheory[mobius](e/d)*p^d ;
%p end do:
%p a := a*f ;
%p end do:
%p a ;
%p end proc:
%p seq(A293303(n),n=1..100) ;
%t s[p_, e_] := DivisorSum[e, MoebiusMu[e/#]*p^#&];
%t a[n_] := a[n] = Times @@ s @@@ FactorInteger[n];
%t Array[a, 100] (* _Jean-François Alcover_, Dec 09 2017 *)
%Y Cf. A000010, A000027, A008683, A166234.
%K nonn,easy,mult
%O 1,2
%A _R. J. Mathar_, Oct 05 2017
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