A300655 -id:A300655 - cp's OEIS frontend

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

Rémy Sigrist, Mar 10 2018

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

			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)

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

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

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

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)

A300669 Positive numbers k with two consecutive ones in the binary representation of 1/k.

Original entry on oeis.org

5, 9, 10, 11, 13, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 29, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 52, 53, 54, 55, 57, 58, 59, 61, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89

Offset: 1

Rémy Sigrist, Mar 11 2018

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 A004780; here 1/k has two consecutive ones in binary, there k has two consecutive ones in binary.
See A300630 for the complementary sequence.

			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      5    0.(0011)
   2      9    0.(0001110)
   3     10    0.0(0011)
   4     11    0.(0001011101)
   5     13    0.0(00100111011)
   6     17    0.(00001111)
   7     18    0.0(000111)
   8     19    0.(000011010111100101)
   9     20    0.00(0011)
  10     21    0.(000011)

Cf. A004780, A300653, A300655, A300630 (complement).

is(n) = my (f=1/max(2, n), s=Set()); while (!setsearch(s, f), if (floor(f*4)==3, return (1), s=setunion(s, Set(f)); f=frac(f*2))); return (0)

cp's OEIS Frontend

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

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