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

1,3

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).

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]

Example

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, ...

Maple

See link.

Crossrefs

Cf. A224763 (denominators), A090822, A224765.

Sequence in context: A309786 A162912 A230070 * A039776 A048864 A379315

Adjacent sequences: A224759 A224760 A224761 * A224763 A224764 A224765

Keyword

nonn,frac

Author

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

Status

approved