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

1, 2, 11, 183, 5871, 375775, 48099263, 12313411455, 6304466665215, 6455773865180671, 13221424875890015231, 54154956291645502388223, 443637401941159955564326911, 7268555193403964711965932118015, 238176016577461115681699663643131903, 15609103422420491677315869156516292427775

Offset

1,2

Comments

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).

Links

Table of n, a(n) for n=1..16.

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

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.

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

Sequence in context: A297601 A089760 A288556 * A295246 A288564 A368561

Adjacent sequences: A353927 A353928 A353929 * A353931 A353932 A353933

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