A300669 -id:A300669 - 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)

A300691 Square array T(n, k) (n >= 1, k >= 1) read by antidiagonals upwards: T(n, k) is the k-th positive number, say m, such that the binary representation of n appears as a substring in the binary representation of 1/m (ignoring the radix point and adding trailing zeros if necessary in case of a terminating expansion).

Original entry on oeis.org

1, 1, 2, 5, 2, 3, 1, 9, 3, 4, 3, 2, 10, 4, 5, 5, 6, 4, 11, 5, 6, 9, 9, 11, 5, 13, 6, 7, 1, 11, 10, 12, 7, 17, 7, 8, 5, 2, 13, 11, 13, 8, 18, 8, 9, 3, 7, 4, 17, 13, 19, 9, 19, 9, 10, 11, 6, 10, 8, 18, 17, 22, 10, 20, 10, 11, 5, 13, 11, 13, 9, 19, 18, 23, 11, 21

Offset: 1

Rémy Sigrist, Mar 11 2018

If m appears in the n-th row, then 2*m also appears in the n-th row.

This array has connections with A300653: here n appears in 1/T(n, k), there T(n, k) appears in 1/n.

			Square array begins:
  n\k|   1   2   3   4   5   6   7   8   9  10  11  12
  ---+------------------------------------------------
    1|   1   2   3   4   5   6   7   8   9  10  11  12  -->  A000027
    2|   1   2   3   4   5   6   7   8   9  10  11  12  -->  A000027
    3|   5   9  10  11  13  17  18  19  20  21  22  23  -->  A300669
    4|   1   2   4   5   7   8   9  10  11  13  14  15
    5|   3   6  11  12  13  19  22  23  24  25  26  27
    6|   5   9  10  11  13  17  18  19  20  21  22  23
    7|   9  11  13  17  18  19  22  25  26  27  29  33
    8|   1   2   4   8   9  11  13  15  16  17  18  19
    9|   5   7  10  13  14  19  20  23  26  27  28  29
   10|   3   6  11  12  19  22  24  25  27  29  35  37
   11|  11  13  19  22  23  25  26  27  29  37  38  43
   12|   5   9  10  13  17  18  19  20  21  23  25  26

Rémy Sigrist, PARI program for A300691

Cf. A000079, A000225, A153055, A300428, A300653, A300669.

PARI
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See Links section.
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T(1, k) = k.
T(2, k) = k.
T(3, k) = A300669(k).
T(n, 1) = A300428(n).
T(n, k) = n for some k iff n belongs to A000079 or to A153055.
T(A000225(i), k) = T(2*A000225(i), k) for any i > 0.

cp's OEIS Frontend

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

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