A056196 Numbers n such that A055229(n) = 2.

8, 24, 32, 40, 56, 72, 88, 96, 104, 120, 128, 136, 152, 160, 168, 184, 200, 224, 232, 248, 264, 280, 288, 296, 312, 328, 344, 352, 360, 376, 384, 392, 408, 416, 424, 440, 456, 472, 480, 488, 504, 512, 520, 536, 544, 552, 568, 584, 600, 608, 616, 632, 640

Offset: 1

Author

Labos Elemer, Aug 02 2000

Keywords

• nonn

Comments

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

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

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

Examples

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

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A036966, A055229, A065465.

Programs

Maple

filter:= proc(n) local T;
    T:= select(t -> (t[2]::odd and t[2]>1), ifactors(n)[2]);
    nops(T) = 1 and T[1][1]=2;
end proc:
select(filter, [$1..1000]); # Robert Israel, Sep 21 2015

Mathematica

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 *)

PARI

isok(n) = my(c=core(n)); gcd(c, n/c) == 2; \\ after PARI in A055229; Michel Marcus, Sep 20 2015

Extensions

Edited by Robert Israel, Sep 21 2015