%I #38 Nov 25 2022 13:24:26

%S 8,24,32,40,56,72,88,96,104,120,128,136,152,160,168,184,200,224,232,

%T 248,264,280,288,296,312,328,344,352,360,376,384,392,408,416,424,440,

%U 456,472,480,488,504,512,520,536,544,552,568,584,600,608,616,632,640

%N Numbers n such that A055229(n) = 2.

%C By definition, the largest square divisor and squarefree part of these numbers have GCD = 2.

%C Different from A036966. E.g., 81 is not here because A055229(81) = 1.

%C Numbers of the form 2^(2*k+1) * m, where k >= 1 and m is an odd number whose prime factorization contains only exponents that are either 1 or even. The asymptotic density of this sequence is (1/12) * Product_{p odd prime} (1-1/(p^2*(p+1))) = A065465 / 11 = 0.08013762179319734335... - _Amiram Eldar_, Dec 04 2020, Nov 25 2022

%H Robert Israel, <a href="/A056196/b056196.txt">Table of n, a(n) for n = 1..10000</a>

%e 88 is here because 88 has squarefree part 22, largest square divisor 4, and A055229(88) = gcd(22, 4) = 2.

%p filter:= proc(n) local T;

%p T:= select(t -> (t[2]::odd and t[2]>1), ifactors(n)[2]);

%p nops(T) = 1 and T[1][1]=2;

%p end proc:

%p select(filter, [$1..1000]); # _Robert Israel_, Sep 21 2015

%t f[n_] := Block[{p = FactorInteger@ n, a}, a = Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 2]} & /@ p); GCD[a, n/a]]; Position[Array[f, 640], 2] // Flatten (* _Michael De Vlieger_, Sep 22 2015, after _Jean-François Alcover_ at A055229 *)

%o (PARI) isok(n) = my(c=core(n)); gcd(c, n/c) == 2; \\ after PARI in A055229; _Michel Marcus_, Sep 20 2015

%Y Cf. A036966, A055229, A065465.

%K nonn

%O 1,1

%A _Labos Elemer_, Aug 02 2000

%E Edited by _Robert Israel_, Sep 21 2015