A224762 -id:A224762 - cp's OEIS frontend

A224765 Denominator of fractional curling number of binary expansion of n.

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

Offset: 0

N. J. A. Sloane, Apr 26 2013

See A224762 for definition and Maple program.

			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.

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

Cf. A224764, A224762, A181935.

A224764 Numerator of fractional curling number of binary expansion of n.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Apr 26 2013

See A224762 for definition and Maple program.

			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.

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

Cf. A224765, A224762, A181935.

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), ...

Original entry on oeis.org

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

Offset: 1

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

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.

			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, ...

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_]

Cf. A224762 (numerators), A090822.

Maple
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See A224762.
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A224765 Denominator of fractional curling number of binary expansion of n.

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A224764 Numerator of fractional curling number of binary expansion of n.

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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), ...

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