A353930 - OEIS

%I #10 Feb 06 2023 08:53:59

%S 1,2,11,183,5871,375775,48099263,12313411455,6304466665215,

%T 6455773865180671,13221424875890015231,54154956291645502388223,

%U 443637401941159955564326911,7268555193403964711965932118015,238176016577461115681699663643131903,15609103422420491677315869156516292427775

%N Smallest number whose binary expansion has n distinct run-sums.

%C Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4).

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

%e The terms, binary expansions, and standard compositions begin:

%e        1:                    1  (1)

%e        2:                   10  (2)

%e       11:                 1011  (2,1,1)

%e      183:             10110111  (2,1,2,1,1,1)

%e     5871:        1011011101111  (2,1,2,1,1,2,1,1,1,1)

%e   375775:  1011011101111011111  (2,1,2,1,1,2,1,1,1,2,1,1,1,1,1)

%t qe=Table[Length[Union[Total/@Split[IntegerDigits[n,2]]]],{n,1,10000}];

%t Table[Position[qe,i][[1,1]],{i,Max@@qe}]

%o (PARI) a(n) = {my(t=1); if(n==2, t<<=1, for(k=3, n, t = (t<<k) + (2^(k-1)-1))); t} \\ _Andrew Howroyd_, Jan 01 2023

%Y Essentially the same as A215203.

%Y For prime indices instead of binary expansion we have A006939.

%Y For lengths instead of sums of runs we have A165933 = firsts in A165413.

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

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

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

%Y These are the positions of first appearances in A353929.

%Y A005811 counts runs in binary expansion.

%Y A242882 counts compositions with distinct multiplicities.

%Y A318928 gives runs-resistance of binary expansion.

%Y A351014 counts distinct runs in standard compositions.

%Y A353835 counts partitions with all distinct run-sums, weak A353861.

%Y A353864 counts rucksack partitions.

%Y Cf. A044813, A073093, A181819, A304442, A353743, A353840, A353841, A353842, A353847, A353848, A353850, A353853, A353932, A354582.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jun 07 2022

%E Offset corrected and terms a(7) and beyond from _Andrew Howroyd_, Jan 01 2023