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A382419 The product of exponents in the prime factorization of the cubefree numbers.

%I A382419 #9 Mar 25 2025 10:11:40
%S A382419 1,1,1,2,1,1,1,2,1,1,2,1,1,1,1,2,1,2,1,1,1,2,1,2,1,1,1,1,1,1,4,1,1,1,
%T A382419 1,1,1,2,2,1,1,2,2,1,2,1,1,1,1,1,2,1,1,2,1,1,1,2,1,1,1,1,1,2,2,1,1,1,
%U A382419 1,1,2,1,1,1,1,2,1,2,1,1,1,1,2,2,4,1,1
%N A382419 The product of exponents in the prime factorization of the cubefree numbers.
%C A382419 Differs from A368712 at n = 1, 31, 85, 151, 164, 189, ... .
%C A382419 All the terms are powers of 2.
%H A382419 Amiram Eldar, <a href="/A382419/b382419.txt">Table of n, a(n) for n = 1..10000</a>
%F A382419 a(n) = A005361(A004709(n)).
%F A382419 a(n) = 2^A376366(n).
%F A382419 a(n) >= A368712(n).
%F A382419 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(3) * Product_{p prime} (1 + 1/p^2 - 2/p^3) = A002117 * A330594 = 1.33062904409568262931... .
%t A382419 s[n_] := Times @@ FactorInteger[n][[;; , 2]]; cubeFreeQ[n_] := Max[FactorInteger[n][[;; , 2]]] < 3; s /@ Select[Range[120], cubeFreeQ]
%o A382419 (PARI) list(kmax) = {my(e); print1(1, ", "); for(k = 2, kmax, e = factor(k)[, 2]; if(vecmax(e) < 3, print1(vecprod(e), ", "))); }
%Y A382419 Cf. A002117, A004709, A005361, A330594, A368712, A376366, A382421, A382422.
%K A382419 nonn,easy
%O A382419 1,4
%A A382419 _Amiram Eldar_, Mar 25 2025