A270000 Harmonic fractility of n.

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

Offset: 2

Author

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

Keywords

• nonn
• easy

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

Examples

			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.
		

Links

• Jack W Grahl, Table of n, a(n) for n = 2..999
• Jack W Grahl, Python code to generate this sequence

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).

Programs

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 *)

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)

Extensions

Definition corrected by Jack W Grahl, Jun 27 2018

Edited by M. F. Hasler, Nov 05 2018