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

Author

Rémy Sigrist, Mar 05 2018

Keywords

• nonn
• base

Comments

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.

Examples

			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...
		

Links

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

Crossrefs

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

Programs

PARI

See Links section.

Formula

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.