A224762 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), ...

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, 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, 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

Offset: 1

Author

Conference dinner party, Workshop on Challenges in Combinatorics on Words, Fields Institute, Toronto, Apr 22 2013, entered by N. J. A. Sloane

Keywords

• nonn
• frac

Comments

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.

For the first 1000 terms, 1 <= S(n) <= 2. Is this always true?

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.

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).

Examples

			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, ...
		

Links

• Allan Wilks, Table of n, a(n) for n = 1..10000 (terms 1..1000 from N. J. A. Sloane)
• N. J. A. Sloane, Maple program for fractional curling number and S(1),S(2),...
• Allan Wilks, Table of n, S(n) for n = 1..10000 [The first 1000 terms were computed by _N. J. A. Sloane_]

Crossrefs

Cf. A224763 (denominators), A090822, A224765.

Programs

Maple

See link.