A300655 a(n) is the length of the longest contiguous block of ones in the binary expansion of 1/n.

1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 3, 1, 3, 1, 1, 1, 4, 3, 4, 2, 2, 3, 2, 1, 4, 3, 4, 1, 4, 1, 1, 1, 5, 4, 3, 3, 5, 4, 2, 2, 5, 2, 5, 3, 2, 2, 3, 1, 4, 4, 1, 3, 5, 4, 4, 1, 5, 4, 5, 1, 5, 1, 1, 1, 6, 5, 6, 4, 3, 3, 3, 3, 3, 5, 3, 4, 4, 2, 4, 2, 6, 5, 6, 2, 2, 5, 4

Offset: 1

Author

Rémy Sigrist, Mar 10 2018

Keywords

• nonn
• base

Comments

This sequence has similarities with A038374: here we consider the binary expansion of 1/n, there the binary expansion of n.

Examples

			The first terms, alongside the binary representation of 1/n, are:
  n   a(n)  bin(1/n) with repeating digits in parentheses
  --  ----  ---------------------------------------------
   1     1  1.(0)
   2     1  0.1(0)
   3     1  0.(01)
   4     1  0.01(0)
   5     2  0.(0011)
   6     1  0.0(01)
   7     1  0.(001)
   8     1  0.001(0)
   9     3  0.(000111)
  10     2  0.0(0011)
  11     3  0.(0001011101)
  12     1  0.00(01)
  13     3  0.(000100111011)
  14     1  0.0(001)
  15     1  0.(0001)
  16     1  0.0001(0)
  17     4  0.(00001111)
  18     3  0.0(000111)
  19     4  0.(000011010111100101)
  20     2  0.00(0011)
		

Crossrefs

Cf. A038374, A300630.

Programs

PARI

a(n) = my (w=1, s=Set(), f=1/max(n,2)); while (!setsearch(s,f), while (floor(f*2^(w+1))==2^(w+1)-1, w++); s=setunion(s,Set(f)); f=frac(f*2)); return (w)

Formula

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

a(2^k + 1) = k for any k > 0.

a(n) = 1 iff n belongs to A300630.