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 A269571 Numbers having binary fractility 1. 3

%I

%S 2,3,4,5,8,11,13,16,19,29,32,37,53,59,61,64,67,83,101,107,128,131,139,

%T 149,163,173,179,181,197,211,227,256,269,293,317,347,349,373,379,389,

%U 419,421,443,461,467,491,509,512,523,541,547,557,563,587,613,619

%N Numbers having binary fractility 1.

%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}.

%H Robert Price, <a href="/A269571/b269571.txt">Table of n, a(n) for n = 1..76</a>

%e D(1/5) = (3,1,3,1,3,1,3,1,...)

%e D(2/5) = (2,1,3,1,3,1,3,1,...)

%e D(3/5) = (1,3,1,3,1,3,1,3,...)

%e D(4/5) = (1,1,3,1,3,1,3,1,...).

%e This shows that all m/5, for 0<m<5 are equivalent to 1/5, so that there is only 1 equivalence class.

%t A269570[n_] := CountDistinct[With[{l = NestWhileList[

%t Rescale[#, {1/2^(Floor[-Log[2, #]] + 1),

%t 1/2^(Floor[-Log[2, #]])}] &, #, UnsameQ, All]},

%t Min@l[[First@First@Position[l, Last@l] ;;]]] & /@

%t Range[1/n, 1 - 1/n, 1/n]] (* from Davin Park, Nov 19 2016 *)

%t Select[Range[1000], A269570[#] == 1 &] (* _Robert Price_, Sep 20 2019 *)

%Y Cf. A269570, A269572.

%K nonn

%O 1,1

%A _Clark Kimberling_, Mar 01 2016

%E Corrected Offset by _Robert Price_, Sep 20 2019

%E a(37)-a(56) from _Robert Price_, Sep 20 2019

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Last modified December 6 18:16 EST 2021. Contains 349567 sequences. (Running on oeis4.)