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

Author

Gus Wiseman, Jun 26 2022

Keywords

• base
• nonn

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.

Examples

			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.
		

Links

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

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.

Programs

Mathematica

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

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