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A368039 The product of exponents of prime factorization of the nonsquarefree numbers.

%I A368039 #7 Dec 09 2023 07:08:58
%S A368039 2,3,2,2,4,2,2,3,2,3,2,5,4,3,2,2,4,2,2,2,3,3,2,2,6,2,6,2,2,4,4,2,3,2,
%T A368039 2,5,2,2,4,3,6,4,2,2,3,2,2,3,2,7,2,3,3,2,8,2,2,2,3,2,2,5,4,2,3,2,2,2,
%U A368039 2,4,4,3,2,3,6,4,2,6,2,2,4,2,9,2,5,4,2
%N A368039 The product of exponents of prime factorization of the nonsquarefree numbers.
%C A368039 The terms of A005361 that are larger than 1, since A005361(k) = 1 if and only if k is squarefree (A005117).
%H A368039 Amiram Eldar, <a href="/A368039/b368039.txt">Table of n, a(n) for n = 1..10000</a>
%F A368039 a(n) = A005361(A013929(n)).
%F A368039 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = ((zeta(2)*zeta(3)/zeta(6)) - 1/zeta(2))/(1-1/zeta(2)) = (A082695 - A059956)/A229099 = 3.406686208821... .
%t A368039 Select[Table[Times @@ FactorInteger[n][[;;, 2]], {n, 1, 250}], # > 1 &]
%o A368039 (PARI) lista(kmax) = {my(p); for(k = 1, kmax, p = vecprod(factor(k)[, 2]); if(p > 1, print1(p, ", ")));}
%Y A368039 Cf. A005117, A005361, A013929.
%Y A368039 Cf. A084936, A174961, A275699, A368038, A368040.
%Y A368039 Cf. A059956, A082695, A229099.
%K A368039 nonn
%O A368039 1,1
%A A368039 _Amiram Eldar_, Dec 09 2023