cp's OEIS Frontend

A353930 Smallest number whose binary expansion has n distinct run-sums.

%I A353930 #10 Feb 06 2023 08:53:59
%S A353930 1,2,11,183,5871,375775,48099263,12313411455,6304466665215,
%T A353930 6455773865180671,13221424875890015231,54154956291645502388223,
%U A353930 443637401941159955564326911,7268555193403964711965932118015,238176016577461115681699663643131903,15609103422420491677315869156516292427775
%N A353930 Smallest number whose binary expansion has n distinct run-sums.
%C A353930 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 A353930 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/q/87559">What is a sequence run? (answered 2011-12-01)</a>
%e A353930 The terms, binary expansions, and standard compositions begin:
%e A353930        1:                    1  (1)
%e A353930        2:                   10  (2)
%e A353930       11:                 1011  (2,1,1)
%e A353930      183:             10110111  (2,1,2,1,1,1)
%e A353930     5871:        1011011101111  (2,1,2,1,1,2,1,1,1,1)
%e A353930   375775:  1011011101111011111  (2,1,2,1,1,2,1,1,1,2,1,1,1,1,1)
%t A353930 qe=Table[Length[Union[Total/@Split[IntegerDigits[n,2]]]],{n,1,10000}];
%t A353930 Table[Position[qe,i][[1,1]],{i,Max@@qe}]
%o A353930 (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 A353930 Essentially the same as A215203.
%Y A353930 For prime indices instead of binary expansion we have A006939.
%Y A353930 For lengths instead of sums of runs we have A165933 = firsts in A165413.
%Y A353930 Numbers whose binary expansion has all distinct runs are A175413.
%Y A353930 For standard compositions we have A246534, firsts of A353849.
%Y A353930 For runs instead of run-sums we have A350952, firsts of A297770.
%Y A353930 These are the positions of first appearances in A353929.
%Y A353930 A005811 counts runs in binary expansion.
%Y A353930 A242882 counts compositions with distinct multiplicities.
%Y A353930 A318928 gives runs-resistance of binary expansion.
%Y A353930 A351014 counts distinct runs in standard compositions.
%Y A353930 A353835 counts partitions with all distinct run-sums, weak A353861.
%Y A353930 A353864 counts rucksack partitions.
%Y A353930 Cf. A044813, A073093, A181819, A304442, A353743, A353840, A353841, A353842, A353847, A353848, A353850, A353853, A353932, A354582.
%K A353930 nonn
%O A353930 1,2
%A A353930 _Gus Wiseman_, Jun 07 2022
%E A353930 Offset corrected and terms a(7) and beyond from _Andrew Howroyd_, Jan 01 2023