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A382890 The square root of the largest square dividing the n-th cubefree number.

%I A382890 #9 Apr 08 2025 13:02:18
%S A382890 1,1,1,2,1,1,1,3,1,1,2,1,1,1,1,3,1,2,1,1,1,5,1,2,1,1,1,1,1,1,6,1,1,1,
%T A382890 1,1,1,2,3,1,1,7,5,1,2,1,1,1,1,1,2,1,1,3,1,1,1,2,1,1,1,1,1,5,2,1,1,1,
%U A382890 1,1,2,1,1,1,1,3,1,2,1,1,1,1,7,3,10,1,1
%N A382890 The square root of the largest square dividing the n-th cubefree number.
%C A382890 The product of the non-unitary prime divisors of the n-th cubefree number.
%C A382890 Also, the square root of the powerful part of the n-th cubefree number.
%C A382890 All the terms are squarefree.
%H A382890 Amiram Eldar, <a href="/A382890/b382890.txt">Table of n, a(n) for n = 1..10000</a>
%F A382890 a(n) = A000188(A004709(n)).
%F A382890 a(n) = sqrt(A382889(n)).
%F A382890 a(n) = A004709(n)/A382888(n).
%F A382890 a(n) = sqrt(A004709(n)/A382891(n)).
%F A382890 a(A371188(n)) = 1.
%t A382890 f[p_, e_] := p^If[e == 1, 0, 1]; s[n_] := Module[{fct = FactorInteger[n]}, If[AllTrue[fct[[;; , 2]], # < 3 &], Times @@ f @@@ fct, Nothing]]; Array[s, 100]
%o A382890 (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, 1)), ", ")));}
%Y A382890 Cf. A000188, A004709, A005117, A057521, A371188 (positions of 1's).
%Y A382890 Similar sequences: A382888, A382889, A382891.
%K A382890 nonn,easy
%O A382890 1,4
%A A382890 _Amiram Eldar_, Apr 07 2025