A181935 Curling number of binary expansion of n.

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

Offset: 0

Author

N. J. A. Sloane, Apr 02 2012

Keywords

• nonn
• base

Comments

Given a string S, write it as S = XYY...Y = XY^k, where X may be empty, and k is as large as possible; then k is the curling number of S.

Examples

			731 = 1011011011 in binary, which we could write as XY^2 with X = 10110110 and Y = 1, or as XY^3 with X = 1 and Y = 011. The latter is better, giving k = 3, so a(713) = 3.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..8191
• Benjamin Chaffin, John P. Linderman, N. J. A. Sloane, and Allan R. Wilks, On Curling Numbers of Integer Sequences, Journal of Integer Sequences, Vol. 16 (2013), Article 13.4.3.
• Index entries for sequences related to binary expansion of n.

Crossrefs

Cf. A212412 (parity), A212439, A212440, A212441, A007088, A090822, A224764/A224765 (fractional curling number).

Programs

Haskell

import Data.List (unfoldr, inits, tails, stripPrefix)
import Data.Maybe (fromJust)
a181935 0 = 1
a181935 n = curling $ unfoldr
   (\x -> if x == 0 then Nothing else Just $ swap $ divMod x 2) n where
   curling zs = maximum $ zipWith (\xs ys -> strip 1 xs ys)
                          (tail $ inits zs) (tail $ tails zs) where
      strip i us vs | vs' == Nothing = i
                    | otherwise      = strip (i + 1) us $ fromJust vs'
                    where vs' = stripPrefix us vs
-- Reinhard Zumkeller, May 16 2012

Mathematica

f[n_, e_] := Module[{d = IntegerDigits[n, 2^e]}, Length[Split[d][[-1]]] - If[SameQ @@ d && Mod[n, 2^e] < 2^(e-1), 1, 0]]; a[n_] := Max[Table[f[n, e], {e, Range[Max[1, Floor[Log2[n]]]]}]]; a[0] = 1; Array[a, 100, 0] (* Amiram Eldar, Apr 08 2025 *)

Formula

A212439(n) = 2*n + a(n) mod 2. - Reinhard Zumkeller, May 17 2012