OFFSET
1,1
COMMENTS
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 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.
In the case of harmonic fractility, r(n) = 1/n, we have n(j+1) = floor(L(j)/(x -Sum_{i=1..j} L(i-1)/(n(i)+1))) for j >= 0, L(0) = 1. - M. F. Hasler, Nov 05 2018
LINKS
Jack W Grahl, Table of n, a(n) for n = 1..299
EXAMPLE
Nested interval sequences NI(k/m) for m = 11:
NI(1/11) = (11,1, 1, 1, 1, 1, 1, 1, ...),
NI(2/11) = (5, 2, 1, 2, 1, 2, 1, 1, 2, ...),
NI(3/11) = (3, 3, 3, 3, 3, 3, 3, 3, 3, ...),
NI(4/11) = (2, 5, 2, 1, 2, 1, 2, 1, 2, ...),
NI(5/11) = (2, 1, 2, 1, 2, 1, 2, 1, 2, ...) equivalent to NI(4/11),
NI(6/11) = (1, 11, 1, 1, 1, 1, 1, 1, ...) equivalent to NI(1/11),
NI(7/11) = (1, 3, 3, 3, 3, 3, 3, 3, 3, ...) equivalent to NI(3/11),
NI(8/11) = (1, 2, 1, 2, 1, 2, 1, 2, 1, ...) equivalent to NI(4/11),
NI(9/11) = (1, 1, 3, 3, 3, 3, 3, 3, 3, ...) equivalent to NI(3/11),
NI(10/11) = (1, 1, 1, 3, 3, 3, 3, 3, ...) equivalent to NI(3/11).
So there are 3 equivalence classes for m = 11, and the fractility of 11 is 3.
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