A300867 a(n) is the least positive k such that k * n is a Fibbinary number (A003714).

1, 1, 1, 3, 1, 1, 3, 3, 1, 1, 1, 3, 3, 5, 3, 11, 1, 1, 1, 7, 1, 1, 3, 3, 3, 13, 5, 3, 3, 5, 11, 11, 1, 1, 1, 39, 1, 1, 7, 7, 1, 1, 1, 3, 3, 13, 3, 7, 3, 21, 13, 23, 5, 5, 3, 3, 3, 9, 5, 11, 11, 9, 11, 43, 1, 1, 1, 35, 1, 1, 39, 15, 1, 1, 1, 31, 7, 57, 7, 7, 1

Offset: 0

Author

Rémy Sigrist, Mar 14 2018

Keywords

• nonn
• look
• base

Comments

This sequence is well defined: for any positive n, according to the pigeonhole principle, A195156(i) mod n = A195156(j) mod n for some distinct i and j, hence n divides f = abs(A195156(i) - A195156(j)), and as f is a Fibbinary number, a(n) <= f/n.

All terms are odd.

Examples

			The first terms, alongside the binary representation of n * a(n), are:
  n  a(n)   bin(n * a(n))
  -- ----   -------------
   0    1           0
   1    1           1
   2    1          10
   3    3        1001
   4    1         100
   5    1         101
   6    3       10010
   7    3       10101
   8    1        1000
   9    1        1001
  10    1        1010
  11    3      100001
  12    3      100100
  13    5     1000001
  14    3      101010
  15   11    10100101
  16    1       10000
  17    1       10001
  18    1       10010
  19    7    10000101
  20    1       10100
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..10000
• Rémy Sigrist, Colored logarithmic scatterplot of the first 1000000 terms (where the color is function of A070939(n * a(n)))

Crossrefs

Cf. A003714, A070939, A195156, A300889.

Programs

PARI

a(n) = forstep (k=1, oo, 2, if (bitand(k*n, 2*k*n)==0, return (k)))

Formula

a(n) = A300889(n) / n for any n > 0.

a(2*n) = a(n).

a(n) = 1 iff n belongs to A003714.