A300475 -id:A300475 - cp's OEIS frontend

A300428 a(n) is the least positive k such that the binary representation of n appears as a substring in the binary representation of 1/k (ignoring the radix point and adding trailing zeros if necessary in case of a terminating expansion).

1, 1, 5, 1, 3, 5, 9, 1, 5, 3, 11, 5, 11, 9, 17, 1, 9, 7, 5, 11, 3, 13, 11, 9, 5, 11, 13, 9, 11, 17, 33, 1, 17, 11, 9, 7, 19, 5, 13, 11, 29, 3, 19, 13, 27, 11, 19, 17, 9, 19, 5, 11, 19, 13, 29, 9, 19, 11, 13, 17, 25, 33, 65, 1, 33, 23, 17, 13, 11, 29, 9, 23, 7

Offset: 1

Rémy Sigrist, Mar 05 2018

In other words, a(n) is the least k > 0 such that floor((2^i) / k) mod A062383(n) = n for some integer i >= 0.
This sequence is similar to A035335 for the base 2.
All terms are odd.
All terms appears infinitely many times (as a(n) equals at least a(2*n) or a(2*n + 1)).
See also A300475 for a similar sequence.

			The first terms, alongside the binary representation of 1/a(n) with the earliest occurrence of the binary representation of n in parentheses, are:
  n  a(n)    bin(1/a(n))
  -- ----    -----------
   1    1    (1).000...
   2    1    (1.0)000...
   3    5    0.00(11)001...
   4    1    (1.00)000...
   5    3    0.0(101)010...
   6    5    0.00(110)011...
   7    9    0.000(111)000...
   8    1    (1.000)000...
   9    5    0.001(1001)100...
  10    3    0.0(1010)101...
  11   11    0.000(1011)101...
  12    5    0.00(1100)110...
  13   11    0.000101(1101)000...
  14    9    0.000(1110)001...
  15   17    0.0000(1111)000...
  16    1    (1.0000)000...
  17    9    0.00011(10001)110...
  18    7    0.00(10010)010...
  19    5    0.001(10011)001...
  20   11    0.0001011(10100)010...

Rémy Sigrist, Table of n, a(n) for n = 1..8191
Rémy Sigrist, PARI program for A300428

Cf. A000975, A033138, A035335, A062383, A300475.

PARI
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See Links section.
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a(2^k) = 1 for any k >= 0.
a(2^k - 1) = 2^k + 1 for any k > 1.
a(A000975(k)) = 3 for any k > 2.
a(A033138(k)) = 7 for any k > 4.
a(n) <= A300475(n) for any n > 0.

A309668 a(n) is the least positive number of the form floor(2^k/n) for some k >= 0 not yet in the sequence.

Original entry on oeis.org

1, 2, 5, 4, 3, 10, 9, 8, 7, 6, 11, 21, 19, 18, 17, 16, 15, 14, 13, 12, 24, 23, 22, 42, 20, 39, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 52, 25, 49, 48, 47, 46, 45, 44, 43, 85, 41, 40, 80, 78, 38, 75, 74, 73, 71, 70, 69, 68, 67, 66, 65, 64, 63, 62, 61

Offset: 1

Rémy Sigrist, Aug 11 2019

The sequence is well defined as for any n > 0, there are infinitely many positive numbers of the form floor(2^k/n) with k >= 0.
The sequence is a permutation of the natural numbers, with inverse A309734:
- for any m > 0, floor(2^k/A300475(m)) = m for some k,
- also, for any u > 0, floor(2^(k-u)/(A300475(m)*2^u)) = m,
- so the set S_m = { v such that floor(2^k/v) = m for some k >= 0 } is infinite
- and eventually a(n) = m for some n in S_m, QED.

			The first terms, alongside the binary representations of a(n) and of 1/n (with that of a(n) in parentheses), are:
  --  ----  ---------  ---------------------
   1     1          1  (1).00000000000000...
   2     2         10  0.(10)000000000000...
   3     5        101  0.0(101)0101010101...
   4     4        100  0.0(100)0000000000...
   5     3         11  0.00(11)0011001100...
   6    10       1010  0.00(1010)10101010...
   7     9       1001  0.00(1001)00100100...
   8     8       1000  0.00(1000)00000000...
   9     7        111  0.000(111)00011100...
  10     6        110  0.000(110)01100110...
  11    11       1011  0.000(1011)1010001...
  12    21      10101  0.000(10101)010101...
  13    19      10011  0.000(10011)101100...
  14    18      10010  0.000(10010)010010...
  15    17      10001  0.000(10001)000100...

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Colored representation of the first 2^16 terms (where the color is function of the parity of A070939(n) - A070939(a(n)))
Index entries for sequences that are permutations of the natural numbers

Cf. A070939, A300475, A309734 (inverse).

s=1; for (n=1, 67, q=1/n; while (bittest(s, f=floor(q)), q*=2); print1 (f ", "); s+=2^f)

cp's OEIS Frontend

A300428 a(n) is the least positive k such that the binary representation of n appears as a substring in the binary representation of 1/k (ignoring the radix point and adding trailing zeros if necessary in case of a terminating expansion).

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