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

1,4

Comments

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

Links

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

Formula

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

Example

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.

Mathematica

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

Prog

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

Crossrefs

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

Cf. A005811, A165414.

Cf. A030308, A044813.

Sequence in context: A022921 A080763 A245920 * A172155 A080573 A186440

Adjacent sequences: A165410 A165411 A165412 * A165414 A165415 A165416

Keyword

base,nonn

Author

Leroy Quet, Sep 17 2009

Extensions

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

Status

approved