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A369429 Square root of the largest square dividing the n-th powerful number.

%I A369429 #8 Jan 23 2024 08:50:09
%S A369429 1,2,2,3,4,5,3,4,6,7,8,6,9,10,6,11,5,8,12,13,14,10,6,15,9,16,12,17,18,
%T A369429 7,19,14,20,12,21,22,10,16,23,24,25,18,15,26,27,28,20,29,12,30,31,22,
%U A369429 18,10,32,33,15,24,34,35,36,21,11,26,37,14,38,39,28,40
%N A369429 Square root of the largest square dividing the n-th powerful number.
%H A369429 Amiram Eldar, <a href="/A369429/b369429.txt">Table of n, a(n) for n = 1..10000</a>
%F A369429 a(n) = A000188(A001694(n)).
%F A369429 a(n) > 1 for n >= 2.
%F A369429 Sum_{k=1..n} a(k) ~ c * n^2, where c = (15/(2*Pi^2))*(zeta(3)/zeta(3/2))^2 = 0.160894210785... .
%t A369429 f[p_, e_] := p^Floor[e/2]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; s /@ Select[Range[2000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &]
%t A369429 (* or *)
%t A369429 f[p_, e_] := p^Floor[e/2]; f[p_, 1] := 0; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[s, 2000], # > 0 &]
%o A369429 (PARI) s(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] == 1, 0, f[i,1]^(f[i,2]\2)));}
%o A369429 lista(kmax) = {my(s1); for(k = 1, kmax, s1 = s(k); if(s1 > 0, print1(s1, ", ")));}
%Y A369429 Cf. A000188, A001694, A087050, A369430.
%Y A369429 Cf. A002117, A078434.
%K A369429 nonn,easy
%O A369429 1,2
%A A369429 _Amiram Eldar_, Jan 23 2024