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

2,4

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.

Links

Table of n, a(n) for n=2..81.

Example

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.

Sequence in context: A226517 A185214 A241719 * A029198 A029175 A186994

Adjacent sequences: A269569 A269570 A269571 * A269573 A269574 A269575

Keyword

nonn,easy

Author

Clark Kimberling, Mar 01 2016

Status

approved