A224762 -id:A224762

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A224763

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 denominators of S(1), S(2), ...

+0
2

1, 1, 1, 1, 2, 1, 2, 1, 5, 1, 4, 1, 2, 3, 1, 3, 2, 4, 7, 1, 5, 12, 1, 3, 4, 7, 7, 1, 5, 4, 5, 2, 15, 1, 6, 1, 2, 5, 7, 13, 1, 5, 4, 13, 1, 4, 3, 23, 1, 4, 2, 14, 1, 4, 2, 4, 1, 4, 2, 4, 1, 53, 1, 6, 29, 1, 3, 20, 1, 3, 3, 1, 3, 3, 1, 14, 24, 1, 6, 15, 1, 3, 9, 1, 3, 3, 4, 29, 1, 5, 24, 14, 16, 1, 5, 5, 1, 3, 13, 1, 3, 3, 16

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,5

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?

See A224762 for definition and Maple program.

LINKS

Allan Wilks, Table of n, a(n) for n = 1..10000 (terms 1..1000 from N. J. A. Sloane)

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 A224762.

CROSSREFS

Cf. A224762 (numerators), A090822.

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

A224765

Denominator of fractional curling number of binary expansion of n.

+0
4

1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 2, 3, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 5, 2, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 6, 2, 1, 1, 1, 5, 1, 1, 3, 2, 1, 1, 3, 5, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,6

COMMENTS

See A224762 for definition and Maple program.

LINKS

N. J. A. Sloane, Table of n, a(n) for n = 0..10000

EXAMPLE

1, 1, 1, 2, 2, 3/2, 1, 3, 3, 4/3, 2, 2, 2, 3/2, 1, 4, 4, 5/4, 5/3, 2, 2, 5/2, 5/3, 3, 3, 4/3, 2, 2, 2, 3/2, 1, 5, 5, 6/5, 3/2, 2, 2, 2, 3/2, 3, 3, 5/3, 3, 2, 2, 2, 3/2, 4, 4, 5/4, 5/3, 2, 2, 5/2, 2, 3, 3, 4/3, 2, 2, 2, 3/2, 1, 6, 6, 7/6, 3/2, 2, ...

For example, 18 = 10010 in binary has fractional curling number 5/4.

CROSSREFS

Cf. A224764, A224762, A181935.

KEYWORD

nonn,frac

AUTHOR

N. J. A. Sloane, Apr 26 2013

STATUS

approved

A224764

Numerator of fractional curling number of binary expansion of n.

+0
3

1, 1, 1, 2, 2, 3, 1, 3, 3, 4, 2, 2, 2, 3, 1, 4, 4, 5, 5, 2, 2, 5, 5, 3, 3, 4, 2, 2, 2, 3, 1, 5, 5, 6, 3, 2, 2, 2, 3, 3, 3, 5, 3, 2, 2, 2, 3, 4, 4, 5, 5, 2, 2, 5, 2, 3, 3, 4, 2, 2, 2, 3, 1, 6, 6, 7, 3, 2, 2, 2, 7, 3, 3, 7, 5, 2, 2, 5, 7, 4

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

See A224762 for definition and Maple program.

LINKS

N. J. A. Sloane, Table of n, a(n) for n = 0..10000

EXAMPLE

For example, 18 = 10010 in binary has fractional curling number 5/4.

CROSSREFS

Cf. A224765, A224762, A181935.

KEYWORD

nonn,frac

AUTHOR

N. J. A. Sloane, Apr 26 2013

STATUS

approved

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