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%I #7 Nov 24 2016 16:40:56
%S 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,
%T 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,
%U 1,6,5,33,4,11,7,21,3,6,18,11,9,15,8,22,2,27
%N Maximal period-length associated with binary fractility of n.
%C 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.
%e n classes a(n)
%e 2 (1) 1
%e 3 (2) 1
%e 4 (1) 1
%e 5 (1,3) 2
%e 6 (1), (2) 1
%e 7 (1,2), (3) 2
%e 8 (1) 1
%e 9 (1), (1,1,4) 3
%e 10 (1), (1,3) 1
%Y Cf. A269570, A269571.
%K nonn,easy
%O 2,4
%A _Clark Kimberling_, Mar 01 2016
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