A353929 Number of distinct sums of runs (of 0's or 1's) in the binary expansion of n.

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

Offset

0,3

Comments

Assuming the binary digits are not all 1, this is one more than the number of different lengths of runs of 1's in the binary expansion of n.

Links

Table of n, a(n) for n=0..86.

Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

Example

The binary expansion of 183 is (1,0,1,1,0,1,1,1), with runs (1), (0), (1,1), (0), (1,1,1), with sums 1, 0, 2, 0, 3, of which four are distinct, so a(183) = 4.

Mathematica

Table[Length[Union[Total/@Split[IntegerDigits[n, 2]]]], {n, 0, 100}]

Prog

(Python)
from itertools import groupby
def A353929(n): return len(set(sum(map(int, y[1])) for y in groupby(bin(n)[2:]))) # Chai Wah Wu, Jun 26 2022

Crossrefs

For lengths of all runs we have A165413, firsts A165933.

Numbers whose binary expansion has distinct runs are A175413.

For runs instead of run-sums we have A297770, firsts A350952.

For prime indices we have A353835, weak A353861, firsts A006939.

For standard compositions we have A353849, firsts A246534.

Positions of first appearances are A353930.

A005811 counts runs in binary expansion.

A044813 lists numbers with distinct run-lengths in binary expansion.

A318928 gives runs-resistance of binary expansion.

A351014 counts distinct runs in standard compositions.

Cf. A215203, A353743, A353832, A353847, A353932, A354579.

Sequence in context: A323406 A160520 A235708 * A297770 A330617 A343240

Adjacent sequences: A353926 A353927 A353928 * A353930 A353931 A353932

Keyword

base,nonn

Author

Gus Wiseman, Jun 26 2022

Status

approved