%I #53 Feb 27 2018 03:33:20
%S 1,1,2,1,3,1,3,2,6,1,5,1,3,4,1,4,3,5,8,1,6,13,1,4,5,8,9,1,6,5,6,3,16,
%T 1,7,1,3,6,8,14,1,6,5,16,1,5,4,24,1,5,3,15,1,5,3,7,1,5,3,7,2,54,1,7,
%U 31,1,4,21,1,4,5,1,4,5,2,15,25,1,7,17,1,4,11,1,4,5,5,30,1,6,25,15,17,1,6,7,1,4,15,1,4,5,19
%N Define a sequence of rationals by S(1)=1; for n>=1, write S(1),...,S(n) as XY^k, Y nonempty, where the fractional exponent k is maximized, and set S(n+1)=k; sequence gives numerators of S(1), S(2), ...
%C k is the "fractional curling number" of S(1),...,S(n). The infinite sequence S(1), S(2), ... is a fractional analog of Gijswijt's sequence A090822.
%C For the first 1000 terms, 1 <= S(n) <= 2. Is this always true?
%C The fractional curling number k of S = (S(1), S(2), ..., S(n)) is defined as follows. Write S = X Y Y ... Y Y' where X may be empty, Y is nonempty, there are say i copies of Y, and Y' is a prefix of Y. There may be many ways to do this. Choose the version in which the ratio k = (i|Y|+|Y'|)/|Y| is maximized; this k is the fractional curling number of S.
%C For example, if S = (S(1), ..., S(6)) = (1, 1, 2, 1, 3/2, 1), the best choice is to take X = 1,1,2, Y = 1,3/2, Y' = 1, giving k = (2+1)/2 = 3/2 = S(7).
%H Allan Wilks, <a href="/A224762/b224762.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from N. J. A. Sloane)
%H N. J. A. Sloane, <a href="/A224762/a224762_1.txt">Maple program for fractional curling number and S(1),S(2),...</a>
%H Allan Wilks, <a href="/A224762/a224762.txt">Table of n, S(n) for n = 1..10000</a> [The first 1000 terms were computed by _N. J. A. Sloane_]
%e The sequence S(1), S(2), ... begins 1, 1, 2, 1, 3/2, 1, 3/2, 2, 6/5, 1, 5/4, 1, 3/2, 4/3, 1, 4/3, 3/2, 5/4, 8/7, 1, 6/5, 13/12, 1, 4/3, 5/4, 8/7, 9/7, 1, 6/5, 5/4, 6/5, 3/2, 16/15, 1, 7/6, 1, 3/2, 6/5, 8/7, 14/13, 1, 6/5, 5/4, 16/13, 1, 5/4, 4/3, 24/23, 1, 5/4, 3/2, 15/14, 1, 5/4, 3/2, 7/4, ...
%p See link.
%Y Cf. A224763 (denominators), A090822, A224765.
%K nonn,frac
%O 1,3
%A Conference dinner party, Workshop on Challenges in Combinatorics on Words, Fields Institute, Toronto, Apr 22 2013, entered by _N. J. A. Sloane_