A300475 a(n) is the least positive k such that the binary representation n appears in front of the binary representation of 1/k (ignoring the radix point and the leading zeros and adding trailing zeros if necessary in case of a terminating expansion).

1, 1, 5, 1, 3, 5, 9, 1, 7, 3, 11, 5, 19, 9, 17, 1, 15, 7, 13, 25, 3, 23, 11, 21, 5, 19, 37, 9, 35, 17, 33, 1, 31, 15, 29, 7, 27, 53, 13, 25, 49, 3, 47, 23, 45, 11, 43, 21, 41, 81, 5, 39, 19, 75, 37, 9, 71, 35, 69, 17, 67, 33, 65, 1, 63, 31, 61, 15, 59, 29, 57

Offset: 1

Author

Rémy Sigrist, Mar 06 2018

Keywords

• nonn
• look
• base

Comments

In other words, a(n) is the least k > 0 such that floor((2^i) / k) = n for some integer i >= 0.

This sequence is similar to A095156 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 A300428 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    7    0.00(1001)001...
  10    3    0.0(1010)101...
  11   11    0.000(1011)101...
  12    5    0.00(1100)110...
  13   19    0.0000(1101)011...
  14    9    0.000(1110)001...
  15   17    0.0000(1111)000...
  16    1    (1.0000)000...
  17   15    0.000(10001)000...
  18    7    0.00(10010)010...
  19   13    0.000(10011)101...
  20   25    0.0000(10100)011...
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, PARI program for A300475
• Rémy Sigrist, Colored logarithmic scatterplot of the first 1000000 terms (where the color is function of A070939(n * a(n)))

Crossrefs

Cf. A000975, A033138, A070939, A095156, A300428.

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) >= A300428(n).