%I #15 Jun 27 2022 01:08:59

%S 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,

%T 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,

%U 2,2,3,3,2,2,2,3,3,3,3,3,2,2,2,3,2,2,3

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

%C 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.

%H Mathematics Stack Exchange, <a href="https://math.stackexchange.com/q/87559">What is a sequence run? (answered 2011-12-01)</a>

%e 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.

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

%o (Python)

%o from itertools import groupby

%o def A353929(n): return len(set(sum(map(int,y[1])) for y in groupby(bin(n)[2:]))) # _Chai Wah Wu_, Jun 26 2022

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

%Y Numbers whose binary expansion has distinct runs are A175413.

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

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

%Y For standard compositions we have A353849, firsts A246534.

%Y Positions of first appearances are A353930.

%Y A005811 counts runs in binary expansion.

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

%Y A318928 gives runs-resistance of binary expansion.

%Y A351014 counts distinct runs in standard compositions.

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

%K base,nonn

%O 0,3

%A _Gus Wiseman_, Jun 26 2022