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A269571 Numbers having binary fractility 1. 3
2, 3, 4, 5, 8, 11, 13, 16, 19, 29, 32, 37, 53, 59, 61, 64, 67, 83, 101, 107, 128, 131, 139, 149, 163, 173, 179, 181, 197, 211, 227, 256, 269, 293, 317, 347, 349, 373, 379, 389, 419, 421, 443, 461, 467, 491, 509, 512, 523, 541, 547, 557, 563, 587, 613, 619 (list; graph; refs; listen; history; text; internal format)
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

Cf. A269570, A269572.

Sequence in context: A186041 A322531 A302591 * A281493 A181341 A174181

Adjacent sequences:  A269568 A269569 A269570 * A269572 A269573 A269574

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

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Last modified October 18 08:10 EDT 2021. Contains 348066 sequences. (Running on oeis4.)