A300630 Positive numbers k without two consecutive ones in the binary representation of 1/k.

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, 112, 120, 124, 126, 127, 128, 192, 195, 204, 224, 240, 248, 252, 254, 255, 256, 384, 390, 399, 408, 448, 451, 455, 480, 496, 504, 508, 510, 511, 512, 768, 771, 775

Offset: 1

Author

Rémy Sigrist, Mar 10 2018

Keywords

• nonn
• base

Comments

Equivalently, these are the numbers k such that A300655(k) = 1.

Equivalently, these are the numbers k such that A300653(k, 3) > 3.

If n belongs to this sequence then 2*n belongs to this sequence.

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.

For any odd term k, there is at least one positive Fibbinary number, say f, such that k * f belongs to A000225.

Apparently, the only Fibbinary numbers that belong to this sequence are the powers of 2 (A000079).

See A300669 for the complementary sequence.

Includes 2^k-1 for all k>=1. - Robert Israel, Jun 27 2018

Examples

			The first terms, alongside the binary representation of 1/a(n), are:
  n   a(n)    bin(1/a(n)) with repeating digits in parentheses
  --  ----    ------------------------------------------------
   1     1    1.(0)
   2     2    0.1(0)
   3     3    0.(01)
   4     4    0.01(0)
   5     6    0.0(01)
   6     7    0.(001)
   7     8    0.001(0)
   8    12    0.00(01)
   9    14    0.0(001)
  10    15    0.(0001)
  11    16    0.0001(0)
  12    24    0.000(01)
  13    28    0.00(001)
  14    30    0.0(0001)
  15    31    0.(00001)
  16    32    0.00001(0)
  17    48    0.0000(01)
  18    51    0.(00000101)
  19    56    0.000(001)
  20    60    0.00(0001)
		

Links

• Robert Israel, Table of n, a(n) for n = 1..629

Crossrefs

Cf. A000079, A000225, A003714, A300653, A300655, A300669.

Programs

Maple

filter:= proc(n) local m,d,r;
  m:= n/2^padic:-ordp(n,2);
  d:= numtheory:-order(2,m);
  r:=(2^d-1)/m;
  Bits:-Or(r,2*r)=3*r
end proc:
select(filter, [$1..1000]); # Robert Israel, Jun 27 2018

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)