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%I #13 Jun 28 2018 04:50:16
%S 1,2,3,4,6,7,8,12,14,15,16,24,28,30,31,32,48,51,56,60,62,63,64,96,102,
%T 112,120,124,126,127,128,192,195,204,224,240,248,252,254,255,256,384,
%U 390,399,408,448,451,455,480,496,504,508,510,511,512,768,771,775
%N Positive numbers k without two consecutive ones in the binary representation of 1/k.
%C Equivalently, these are the numbers k such that A300655(k) = 1.
%C Equivalently, these are the numbers k such that A300653(k, 3) > 3.
%C If n belongs to this sequence then 2*n belongs to this sequence.
%C This sequence has similarities with the Fibbinary numbers (A003714); here 1/k has no two consecutive ones in binary, there k has no two consecutive ones in binary.
%C For any odd term k, there is at least one positive Fibbinary number, say f, such that k * f belongs to A000225.
%C Apparently, the only Fibbinary numbers that belong to this sequence are the powers of 2 (A000079).
%C See A300669 for the complementary sequence.
%C Includes 2^k-1 for all k>=1. - _Robert Israel_, Jun 27 2018
%H Robert Israel, <a href="/A300630/b300630.txt">Table of n, a(n) for n = 1..629</a>
%e The first terms, alongside the binary representation of 1/a(n), are:
%e n a(n) bin(1/a(n)) with repeating digits in parentheses
%e -- ---- ------------------------------------------------
%e 1 1 1.(0)
%e 2 2 0.1(0)
%e 3 3 0.(01)
%e 4 4 0.01(0)
%e 5 6 0.0(01)
%e 6 7 0.(001)
%e 7 8 0.001(0)
%e 8 12 0.00(01)
%e 9 14 0.0(001)
%e 10 15 0.(0001)
%e 11 16 0.0001(0)
%e 12 24 0.000(01)
%e 13 28 0.00(001)
%e 14 30 0.0(0001)
%e 15 31 0.(00001)
%e 16 32 0.00001(0)
%e 17 48 0.0000(01)
%e 18 51 0.(00000101)
%e 19 56 0.000(001)
%e 20 60 0.00(0001)
%p filter:= proc(n) local m,d,r;
%p m:= n/2^padic:-ordp(n,2);
%p d:= numtheory:-order(2,m);
%p r:=(2^d-1)/m;
%p Bits:-Or(r,2*r)=3*r
%p end proc:
%p select(filter, [$1..1000]); # _Robert Israel_, Jun 27 2018
%o (PARI) is(n) = my (f=1/max(2,n), s=Set()); while (!setsearch(s, f), if (floor(f*4)==3, return (0), s=setunion(s,Set(f)); f=frac(f*2))); return (1)
%Y Cf. A000079, A000225, A003714, A300653, A300655, A300669.
%K nonn,base
%O 1,2
%A _Rémy Sigrist_, Mar 10 2018
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