OFFSET
1,1
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 largest index n such that 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 equal (up to an offset). 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 = 1..117
Peter J. C. Moses, Clark Kimberling, Nested interval sequences of positive real numbers, Integers 17 (2017), #A46.
EXAMPLE
Nested-interval sequences NI(k/m) for m = 6:
NI(1/6) = (6, 1, 1, 1, 1, 1,...)
NI(2/6) = (3, 1, 1, 1, 1, 1,...)
NI(3/6) = (2, 1, 1, 1, 1, 1,...)
NI(4/6) = (1, 3, 1, 1, 1, 1,...)
NI(5/6) = (1, 1, 3, 1, 1, 1,...):
There is only one equivalence class, so that the fractility of 6 is 1.
MATHEMATICA
(* see A270000 *)
PROG
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling and Peter J. C. Moses, Mar 05 2016
EXTENSIONS
Edited by M. F. Hasler, Nov 05 2018
STATUS
approved