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A382891 The powerfree part of the n-th cubefree number.

%I A382891 #9 Apr 08 2025 12:23:22
%S A382891 1,2,3,1,5,6,7,1,10,11,3,13,14,15,17,2,19,5,21,22,23,1,26,7,29,30,31,
%T A382891 33,34,35,1,37,38,39,41,42,43,11,5,46,47,1,2,51,13,53,55,57,58,59,15,
%U A382891 61,62,7,65,66,67,17,69,70,71,73,74,3,19,77,78,79,82,83
%N A382891 The powerfree part of the n-th cubefree number.
%C A382891 Also, the squarefree part of the n-th cubefree number.
%H A382891 Amiram Eldar, <a href="/A382891/b382891.txt">Table of n, a(n) for n = 1..10000</a>
%F A382891 a(n) = A055231(A004709(n)).
%F A382891 a(n) = A007913(A004709(n)).
%F A382891 a(n) = A004709(n)/A382889(n) = A004709(n)/A382890(n)^2.
%F A382891 a(n) = A382888(n)^2/A004709(n).
%F A382891 a(A371188(n)) = A005117(n).
%F A382891 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(3)^2 * Product_{p prime} (1 - 1/p^2 + 1/p^4 - 1/p^5) = 0.92517253037215590197... .
%t A382891 f[p_, e_] := p^If[e == 1, 1, 0]; s[n_] := Module[{fct = FactorInteger[n]}, If[AllTrue[fct[[;; , 2]], # < 3 &], Times @@ f @@@ fct, Nothing]]; Array[s, 100]
%o A382891 (PARI) list(lim) = {my(f); print1(1, ", "); for(k = 2, lim, f = factor(k); if(vecmax(f[, 2]) < 3, print1(prod(i = 1, #f~, f[i, 1]^if(f[i, 2] == 1, 1, 0)), ", ")));}
%Y A382891 Cf. A002117, A004709, A005117, A055231, A007913, A371188.
%Y A382891 Similar sequences: A382888, A382889, A382890.
%K A382891 nonn,easy
%O A382891 1,2
%A A382891 _Amiram Eldar_, Apr 07 2025