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A270000 Harmonic fractility of n. 16
1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 3, 1, 2, 1, 1, 1, 3, 2, 1, 3, 3, 1, 3, 3, 1, 1, 3, 2, 1, 1, 3, 1, 3, 1, 2, 3, 3, 2, 4, 1, 2, 3, 2, 3, 3, 1, 1, 3, 1, 3, 3, 1, 5, 1, 3, 3, 2, 2, 2, 1, 1, 1, 5, 3, 3, 1, 3, 3, 4, 1, 2, 2, 3, 3, 6, 3, 3, 2, 1, 4, 3, 1, 2, 2, 3, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,4

COMMENTS

In order to define (harmonic) fractility of an integer m > 1, we first define nested interval sequences. Suppose that r = (r(n)) is a sequence satisfying (i) 1 = r(1) > r(2) > r(3) > ... and (ii) r(n) -> 0. For x in (0,1], let n(1) be the index n such that r(n+1) < x <= r(n), and let L(1) = r(n(1)) - r(n(1)+1). Let n(2) be the index n such that r(n(1)+1) + L(1)*r(n+1) < x <= r(n(1)+1) + L(1)*r(n), and let L(2) = (r(n(2))-r(n(2)+1))*L(1). Continue inductively to obtain the sequence (n(1), n(2), n(3), ...) =: NI(x), the r-nested interval sequence of x.

For fixed r, call x and y equivalent if NI(x) and NI(y) are eventually identical. For m > 1, the r-fractility of m is the number of equivalence classes of sequences NI(k/m) for 0 < k < m. Taking r = (1/1, 1/2, 1/3, 1/4, ... ) gives harmonic fractility.

For harmonic fractility, r(n) = 1/n, n(j+1) = floor(L(j)/(x - Sum_{i=1..j} L(i-1)/(n(i)+1))) for all j >= 0, L(0) = 1. - M. F. Hasler, Nov 05 2018

LINKS

Jack W Grahl, Table of n, a(n) for n = 2..999

Jack W Grahl, Python code to generate this sequence

EXAMPLE

NI(1/11) = (11, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...),

NI(2/11) = (5, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, ...),

NI(3/11) = (3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, ...),

NI(4/11) = (2, 5, 2, 1, 2, 1, 2, 1, 2, 1, 2, ...),

NI(5/11) = (2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, ...),

NI(6/11) = (1, 11, 1, 1, 1, 1, 1, 1, 1, 1, ...),

NI(7/11) = (1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, ...),

NI(8/11) = (1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, ...),

NI(9/11) = (1, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, ...),

NI(10/11) = (1, 1, 1, 3, 3, 3, 3, 3, 3, 3, ...),

so that there are 3 equivalence classes for n = 11, and that the harmonic fractility of 11 is 3.

MATHEMATICA

A270000[n_] := CountDistinct[With[{l = NestWhileList[Rescale[#, {1/(Floor[1/#] + 1), 1/Floor[1/#]}] &, #, UnsameQ, All]}, Min@l[[First@FirstPosition[l, Last@l] ;; ]]] & /@ Range[1/n, 1 - 1/n, 1/n]] (* Davin Park, Nov 09 2016 *)

PROG

From M. F. Hasler, Nov 05 2018: (Start)

(PARI) A270000(n)=#Set(vector(n-1, k, NIR(k/n))) \\ where:

NIR(x, n, L=1, S=[], c=0)={for(i=2, oo, n=L\x; S=setunion(S, [x/L]); x-=L/(n+1); L/=n*(n+1); setsearch(S, x/L)&& if(c, break, c=!S=[])); S[1]} \\ variant of the function NI() below; returns just a unique representative (smallest x/L occurring within the period) of the equivalence class.

NI(x, n=[], L=1, S=[], c=0)={for(i=2, oo, n=concat(n, L\x); c|| S=setunion(S, [x/L]); x-=L/(n[#n]+1); L/=n[#n]*(n[#n]+1); if(!c, setsearch(S, x/L)&& [c, S]=[i, x/L], x/L==S, c-=i; break)); [n[1..2*c-1], n[c..-1]]} \\ Returns the harmonic nested interval sequence for x in the form [transition, period]. (End)

CROSSREFS

Guide to related sequences:

  k - numbers with harmonic fractility k:

  1 - A269804

  2 - A269805

  3 - A269806

  4 - A269807

  5 - A269808

  6 - A269809

Cf. A269570 (binary fractility), A269982 (factorial fractility).

Sequence in context: A201913 A232463 A325032 * A029384 A225485 A321913

Adjacent sequences:  A269997 A269998 A269999 * A270001 A270002 A270003

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling and Peter J. C. Moses, Mar 11 2016

EXTENSIONS

Definition corrected by Jack W Grahl, Jun 27 2018

Edited by M. F. Hasler, Nov 05 2018

STATUS

approved

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Last modified May 29 14:18 EDT 2020. Contains 334700 sequences. (Running on oeis4.)