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A382888 The squarefree kernel of the n-th cubefree number.

%I A382888 #9 Apr 08 2025 12:49:59
%S A382888 1,2,3,2,5,6,7,3,10,11,6,13,14,15,17,6,19,10,21,22,23,5,26,14,29,30,
%T A382888 31,33,34,35,6,37,38,39,41,42,43,22,15,46,47,7,10,51,26,53,55,57,58,
%U A382888 59,30,61,62,21,65,66,67,34,69,70,71,73,74,15,38,77,78,79,82
%N A382888 The squarefree kernel of the n-th cubefree number.
%H A382888 Amiram Eldar, <a href="/A382888/b382888.txt">Table of n, a(n) for n = 1..10000</a>
%F A382888 a(n) = A007947(A004709(n)).
%F A382888 a(n) = A004709(n)/sqrt(A382889(n)) = A004709(n)/A382890(n).
%F A382888 a(n) = sqrt(A004709(n)*A382891(n)).
%F A382888 a(A371188(n)) = A005117(n).
%F A382888 Sum_{k=1..n} a(k) ~ c * n^2 / 2. where c = zeta(3)^2 * Product_{p prime} (1 - 1/p^2 + 1/p^3 - 1/p^4) = 0.98875409459226057523... .
%t A382888 s[n_] := Module[{f = FactorInteger[n]}, If[AllTrue[f[[;; , 2]], # < 3 &], Times @@ f[[;; , 1]], Nothing]]; Array[s, 100]
%o A382888 (PARI) list(lim) = {my(f); print1(1, ", "); for(k = 2, lim, f = factor(k); if(vecmax(f[, 2]) < 3, print1(vecprod(f[, 1]), ", ")));}
%Y A382888 Cf. A002117, A004709, A005117, A007947, A371188.
%Y A382888 Similar sequences: A382889, A382890, A382891.
%K A382888 nonn,easy
%O A382888 1,2
%A A382888 _Amiram Eldar_, Apr 07 2025