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

1,3

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.

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)))

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).

Example

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...

Prog

(PARI) See Links section.

Crossrefs

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

Sequence in context: A327968 A213815 A300428 * A143113 A053544 A174879

Adjacent sequences: A300472 A300473 A300474 * A300476 A300477 A300478

Keyword

nonn,look,base

Author

Rémy Sigrist, Mar 06 2018

Status

approved