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A368541 The number of exponential divisors of the nonsquarefree numbers.

%I A368541 #9 Dec 30 2023 09:33:32
%S A368541 2,2,2,2,3,2,2,2,2,2,2,2,4,2,2,2,3,2,2,2,2,2,2,2,4,2,4,2,2,3,3,2,2,2,
%T A368541 2,2,2,2,4,2,4,3,2,2,2,2,2,2,2,2,2,2,2,2,6,2,2,2,2,2,2,2,3,2,2,2,2,2,
%U A368541 2,3,4,2,2,2,4,4,2,4,2,2,3,2,4,2,2,4,2,2,2,2,3,2,2,2,2,2,2,4,4,2,2,2,2,2,3,2,2,2,2,2,4,2,2,2,2,2,4,3
%N A368541 The number of exponential divisors of the nonsquarefree numbers.
%C A368541 The terms of A049419 that are larger than 1, since A049419(k) = 1 if and only if k is squarefree (A005117).
%H A368541 Amiram Eldar, <a href="/A368541/b368541.txt">Table of n, a(n) for n = 1..10000</a>
%F A368541 a(n) = A049419(A013929(n)).
%F A368541 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (A327837 - A059956)/A229099 = 2.53623753427906735929... .
%t A368541 f[p_, e_] := DivisorSigma[0, e]; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[s, 200], # > 1 &]
%o A368541 (PARI) lista(kmax) = {my(p, f); for(k = 1, kmax, f = factor(k); p = prod(i=1, #f~, numdiv(f[i, 2])); if(p > 1, print1(p, ", ")));}
%Y A368541 Cf. A005117, A013929, A049419.
%Y A368541 Cf. A084936, A174961, A275699, A368038, A368039, A368040.
%Y A368541 Cf. A013661, A059956, A229099, A327837.
%K A368541 nonn
%O A368541 1,1
%A A368541 _Amiram Eldar_, Dec 29 2023