A269572 Maximal period-length associated with binary fractility of n.

1, 1, 1, 2, 1, 2, 1, 3, 2, 5, 1, 6, 2, 3, 1, 4, 3, 9, 2, 4, 5, 7, 1, 10, 6, 9, 2, 14, 3, 4, 1, 5, 4, 7, 3, 18, 9, 8, 2, 10, 4, 7, 5, 7, 7, 14, 1, 11, 10, 6, 6, 26, 9, 12, 2, 9, 14, 29, 3, 30, 4, 5, 1, 6, 5, 33, 4, 11, 7, 21, 3, 6, 18, 11, 9, 15, 8, 22, 2, 27

Offset: 2

Author

Clark Kimberling, Mar 01 2016

Keywords

• nonn
• easy

Comments

For each x in (0,1], let 1/2^p(1) + 1/2^p(2) + ... be the infinite binary representation of x. Let d(1) = p(1) and d(i) = p(i) - p(i-1) for i >=2. Call (d(i)) the powerdifference sequence of x, and denote it by D(x). Call m/n and u/v equivalent if every period of D(m/n) is a period of D(u/v). Define the binary fractility of n to be the number of distinct equivalence classes of {m/n: 0 < m < n}. Each class is represented by a minimal period, and a(n) is the length of the longest such period.

Examples

			n        classes          a(n)
2         (1)              1
3         (2)              1
4         (1)              1
5         (1,3)            2
6         (1), (2)         1
7         (1,2), (3)       2
8         (1)              1
9         (1), (1,1,4)     3
10        (1), (1,3)       1
		

Crossrefs

Cf. A269570, A269571.