%I #8 Jan 23 2024 16:30:03

%S 1,2,4,3,4,5,9,8,6,7,8,12,9,10,18,11,25,16,12,13,14,20,36,15,27,16,24,

%T 17,18,49,19,28,20,36,21,22,50,32,23,24,25,36,45,26,27,28,40,29,72,30,

%U 31,44,54,100,32,33,75,48,34,35,36,63,121,52,37,98,38,39

%N Smallest number whose square is divisible by the n-th powerful number.

%H Amiram Eldar, <a href="/A369430/b369430.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A019554(A001694(n)).

%F a(n) > 1 for n >= 2.

%F Sum_{k=1..n} a(k) ~ (c*n/Pi)^2 * (2*log(n) + 2*log(c) + 5*gamma - 1 - 6*zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620), and c = zeta(3)/zeta(3/2).

%t f[p_, e_] := p^Ceiling[e/2]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; s /@ Select[Range[2000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &]

%t (* or *)

%t f[p_, e_] := p^Ceiling[e/2]; f[p_, 1] := 0; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[s, 2000], # > 0 &]

%o (PARI) s(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] == 1, 0, f[i,1]^ceil(f[i,2]/2)));}

%o lista(kmax) = {my(s1); for(k = 1, kmax, s1 = s(k); if(s1 > 0, print1(s1, ", ")));}

%Y Cf. A001694, A019554, A369429.

%Y Cf. A001620, A002117, A078434.

%K nonn,easy

%O 1,2

%A _Amiram Eldar_, Jan 23 2024