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A353930
Smallest number whose binary expansion has n distinct run-sums.
7
1, 2, 11, 183, 5871, 375775, 48099263, 12313411455, 6304466665215, 6455773865180671, 13221424875890015231, 54154956291645502388223, 443637401941159955564326911, 7268555193403964711965932118015, 238176016577461115681699663643131903, 15609103422420491677315869156516292427775
OFFSET
1,2
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).
EXAMPLE
The terms, binary expansions, and standard compositions begin:
1: 1 (1)
2: 10 (2)
11: 1011 (2,1,1)
183: 10110111 (2,1,2,1,1,1)
5871: 1011011101111 (2,1,2,1,1,2,1,1,1,1)
375775: 1011011101111011111 (2,1,2,1,1,2,1,1,1,2,1,1,1,1,1)
MATHEMATICA
qe=Table[Length[Union[Total/@Split[IntegerDigits[n, 2]]]], {n, 1, 10000}];
Table[Position[qe, i][[1, 1]], {i, Max@@qe}]
PROG
(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
CROSSREFS
Essentially the same as A215203.
For prime indices instead of binary expansion we have A006939.
For lengths instead of sums of runs we have A165933 = firsts in A165413.
Numbers whose binary expansion has all distinct runs are A175413.
For standard compositions we have A246534, firsts of A353849.
For runs instead of run-sums we have A350952, firsts of A297770.
These are the positions of first appearances in A353929.
A005811 counts runs in binary expansion.
A242882 counts compositions with distinct multiplicities.
A318928 gives runs-resistance of binary expansion.
A351014 counts distinct runs in standard compositions.
A353835 counts partitions with all distinct run-sums, weak A353861.
A353864 counts rucksack partitions.
Sequence in context: A297601 A089760 A288556 * A295246 A288564 A368561
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 07 2022
EXTENSIONS
Offset corrected and terms a(7) and beyond from Andrew Howroyd, Jan 01 2023
STATUS
approved

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Last modified September 20 11:54 EDT 2024. Contains 376068 sequences. (Running on oeis4.)