A165413 a(n) is the number of distinct lengths of runs in the binary representation of n.

1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 1, 2, 2, 2, 3, 1, 3, 2, 3, 2, 2, 2, 1, 2, 2, 2, 3, 3, 2, 3, 2, 3, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 1, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 3, 2

Offset: 1

Author

Leroy Quet, Sep 17 2009

Keywords

• base
• nonn

Comments

Least k whose value is n: 1, 4, 35, 536, 16775, 1060976, ..., = A165933. - Robert G. Wilson v, Sep 30 2009

Examples

			92 in binary is 1011100. There is a run of one 1, followed by a run of one 0, then a run of three 1's, then finally a run of two 0's. The run lengths are therefore (1,1,3,2). The distinct values of these run lengths are (1,3,2). Since there are 3 distinct values, then a(92) = 3.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A140690 (locations of 1's), A165933 (locations of new highs).

Cf. A005811, A165414.

Cf. A030308, A044813.

Programs

Haskell

import Data.List (group, nub)
a165413 = length . nub . map length . group . a030308_row
-- Reinhard Zumkeller, Mar 02 2013

Mathematica

f[n_] := Length@ Union@ Map[ Length, Split@ IntegerDigits[n, 2]]; Array[f, 105] (* Robert G. Wilson v, Sep 30 2009 *)

PARI

binruns(n) = {
  if (n == 0, return([1, 0]));
  my(bag = List(), v=0);
  while(n != 0,
        v = valuation(n,2); listput(bag, v); n >>= v; n++;
        v = valuation(n,2); listput(bag, v); n >>= v; n--);
  return(Vec(bag));
};
a(n) = #Set(select(k->k, binruns(n)));
vector(105, i, a(i))  \\ Gheorghe Coserea, Sep 17 2015

Python

from itertools import groupby
def a(n): return len(set([len(list(g)) for k, g in groupby(bin(n)[2:])]))
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, Jan 04 2021

Formula

a(n) = 1 for n in A140690. - Robert G. Wilson v, Sep 30 2009

Extensions

More terms from Robert G. Wilson v, Sep 30 2009