The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

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