OFFSET
1,1
COMMENTS
In order to define (harmonic) fractility of an integer n > 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 n > 1, the r-fractility of n is the number of equivalence classes of sequences NI(m/n) for 0 < m < n. Taking r = (1/1, 1/2, 1/3, 1/4, ...) gives harmonic fractility.
LINKS
Jack W Grahl, Table of n, a(n) for n = 1..189
EXAMPLE
Nested interval sequences NI(k/m) for m = 5:
NI(1/5) = (5, 1, 1, 1, 1, 1,...),
NI(2/5) = (2, 2, 2, 2, 2, 2,...),
NI(3/5) = (1, 5, 1, 1, 1, 1,...),
NI(4/5) = (1, 1, 5, 1, 1, 1,...),
so that there are 2 equivalence classes for n = 5, and the fractility of 5 is 2.
PROG
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling and Peter J. C. Moses, Mar 05 2016
EXTENSIONS
Edited by M. F. Hasler, Nov 05 2018
STATUS
approved