OFFSET
1,1
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}.
LINKS
Robert Price, Table of n, a(n) for n = 1..76
EXAMPLE
D(1/5) = (3,1,3,1,3,1,3,1,...)
D(2/5) = (2,1,3,1,3,1,3,1,...)
D(3/5) = (1,3,1,3,1,3,1,3,...)
D(4/5) = (1,1,3,1,3,1,3,1,...).
This shows that all m/5, for 0<m<5 are equivalent to 1/5, so that there is only 1 equivalence class.
MATHEMATICA
A269570[n_] := CountDistinct[With[{l = NestWhileList[
Rescale[#, {1/2^(Floor[-Log[2, #]] + 1),
1/2^(Floor[-Log[2, #]])}] &, #, UnsameQ, All]},
Min@l[[First@First@Position[l, Last@l] ;; ]]] & /@
Range[1/n, 1 - 1/n, 1/n]] (* from Davin Park, Nov 19 2016 *)
Select[Range[1000], A269570[#] == 1 &] (* Robert Price, Sep 20 2019 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Mar 01 2016
EXTENSIONS
Corrected Offset by Robert Price, Sep 20 2019
a(37)-a(56) from Robert Price, Sep 20 2019
STATUS
approved