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A382787 The product of exponents in the prime factorization of the numbers whose prime factorization contains exponents that are either 1 or even.

%I A382787 #7 Apr 05 2025 09:10:42
%S A382787 1,1,1,2,1,1,1,2,1,1,2,1,1,1,4,1,2,1,2,1,1,1,2,1,2,1,1,1,1,1,1,4,1,1,
%T A382787 1,1,1,1,2,2,1,1,4,2,2,1,2,1,1,1,1,1,2,1,1,2,6,1,1,1,2,1,1,1,1,1,2,2,
%U A382787 1,1,1,4,4,1,1,2,1,1,1,1,2,1,2,1,1,1,1
%N A382787 The product of exponents in the prime factorization of the numbers whose prime factorization contains exponents that are either 1 or even.
%C A382787 First differs from A368473 at n = 57.
%H A382787 Amiram Eldar, <a href="/A382787/b382787.txt">Table of n, a(n) for n = 1..10000</a>
%F A382787 a(n) = A005361(A335275(n)).
%F A382787 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (zeta(2)^2 / A065465) *  Product_{p prime} (1 - 1/p^2 - 2/p^3 + 3/p^4 - 1/p^6) = 1.568148713987289233406... .
%t A382787 f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[AllTrue[e, # == 1 || EvenQ[#] &], Times @@ e, Nothing]]; Array[f, 150]
%o A382787 (PARI) list(lim) = {my(e, ok); for(k = 1, lim, e = factor(k)[, 2]; ok = 1; for(i = 1, #e, if(e[i] > 1 && e[i]%2, ok = 0; break)); if(ok, print1(vecprod(e), ", ")));}
%Y A382787 Cf. A005361, A013661, A065465, A335275, A368473.
%K A382787 nonn,easy
%O A382787 1,4
%A A382787 _Amiram Eldar_, Apr 05 2025