cp's OEIS Frontend

A382889 The largest square dividing the n-th cubefree number.

%I A382889 #9 Apr 08 2025 12:23:31
%S A382889 1,1,1,4,1,1,1,9,1,1,4,1,1,1,1,9,1,4,1,1,1,25,1,4,1,1,1,1,1,1,36,1,1,
%T A382889 1,1,1,1,4,9,1,1,49,25,1,4,1,1,1,1,1,4,1,1,9,1,1,1,4,1,1,1,1,1,25,4,1,
%U A382889 1,1,1,1,4,1,1,1,1,9,1,4,1,1,1,1,49,9,100
%N A382889 The largest square dividing the n-th cubefree number.
%C A382889 Also, the powerful part of the n-th cubefree number.
%C A382889 All the terms are squares of squarefree numbers (A062503).
%H A382889 Amiram Eldar, <a href="/A382889/b382889.txt">Table of n, a(n) for n = 1..10000</a>
%F A382889 a(n) = A008833(A004709(n)).
%F A382889 a(n) = A057521(A004709(n)).
%F A382889 a(n) = A382890(n)^2.
%F A382889 a(n) = A004709(n)/A382891(n).
%F A382889 a(n) = (A004709(n)/A382888(n))^2.
%F A382889 a(A371188(n)) = 1.
%F A382889 Sum_{k=1..n} a(k) ~ c * n^(3/2) / 3, where c = zeta(3)^(3/2) * Product_{p prime} (1 + 1/p^(3/2) - 1/p^2 - 1/p^(5/2)) = 1.48513488319516447978... .
%t A382889 f[p_, e_] := p^If[e == 1, 0, 2]; s[n_] := Module[{fct = FactorInteger[n]}, If[AllTrue[fct[[;; , 2]], # < 3 &], Times @@ f @@@ fct, Nothing]]; Array[s, 100]
%o A382889 (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, 0, 2)), ", ")));}
%Y A382889 Cf. A002117, A004709, A008833, A057521, A062503, A371188 (positions of 1's).
%Y A382889 Similar sequences: A382888, A382890, A382891.
%K A382889 nonn,easy
%O A382889 1,4
%A A382889 _Amiram Eldar_, Apr 07 2025