A353840 -id:A353840 - cp's OEIS frontend

A353862 Greatest run-sum of the prime indices of n.

0, 1, 2, 2, 3, 2, 4, 3, 4, 3, 5, 2, 6, 4, 3, 4, 7, 4, 8, 3, 4, 5, 9, 3, 6, 6, 6, 4, 10, 3, 11, 5, 5, 7, 4, 4, 12, 8, 6, 3, 13, 4, 14, 5, 4, 9, 15, 4, 8, 6, 7, 6, 16, 6, 5, 4, 8, 10, 17, 3, 18, 11, 4, 6, 6, 5, 19, 7, 9, 4, 20, 4, 21, 12, 6, 8, 5, 6, 22, 4, 8

Offset: 1

Gus Wiseman, May 23 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

A run-sum of a sequence is the sum of any maximal consecutive constant subsequence.

			The prime indices of 72 are {1,1,1,2,2}, with run-sums {3,4}, so a(72) = 4.

Positions of first appearances are A008578.
For binary expansion we have A038374, least A144790.
For run-lengths instead of run-sums we have A051903.
Distinct run-sums are counted by A353835, weak A353861.
The least run-sum is given by A353931.
A001222 counts prime factors, distinct A001221.
A005811 counts runs in binary expansion.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A300273 ranks collapsible partitions, counted by A275870.
A304442 counts partitions with all equal run-sums, compositions A353851.
A353832 represents the operation of taking run-sums of a partition.
A353833 ranks partitions with all equal run sums, nonprime A353834.
A353838 ranks partitions with all distinct run-sums, counted by A353837.
A353840-A353846 pertain to partition run-sum trajectory.
Cf. A067340, A073093, A181819, A182857, A316413, A353836, A353839, A353848, A353852, A353866.

Table[Max@@Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k],{n,100}]

A353743 Least number with run-sum trajectory of length k; a(0) = 1.

Original entry on oeis.org

1, 2, 4, 12, 84, 1596, 84588, 11081028, 3446199708, 2477817590052, 4011586678294188, 14726534696017964148, 120183249654202605411828, 2146833388573021140471483564, 83453854313999050793547980583372, 7011542477899258250521520684673165324

Offset: 0

Gus Wiseman, Jun 11 2022

nonn

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). The run-sum trajectory is obtained by repeatedly taking the run-sum transformation (A353832, A353847) until a squarefree number is reached. For example, the trajectory 12 -> 9 -> 7 corresponds to the partitions (2,1,1) -> (2,2) -> (4).

			The terms together with their prime indices begin:
      1: {}
      2: {1}
      4: {1,1}
     12: {1,1,2}
     84: {1,1,2,4}
   1596: {1,1,2,4,8}
  84588: {1,1,2,4,8,16}

The ordered version is A072639, for run-lengths A333629.
The version for run-lengths is A325278, firsts in A182850 or A323014.
The run-sum trajectory is the iteration of A353832.
The first length-k row of A353840 has index a(k).
Other sequences pertaining to this trajectory are A353841-A353846.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A300273 ranks collapsible partitions, counted by A275870.
A353833 ranks partitions with all equal run-sums, counted by A304442.
A353835 counts distinct run-sums of prime indices, weak A353861.
A353838 ranks partitions with all distinct run-sums, counted by A353837.
A353866 ranks rucksack partitions, counted by A353864.
Cf. A002033, A005117, A006939, A071625, A076954, A126796, A181819, A182857, A188431, A299702, A325780, A325781, A353834.

Join[{1,2},Table[2*Product[Prime[2^k],{k,0,n}],{n,0,6}]]

a(n > 1) = 2 * Product_{k=0..n-2} prime(2^k).

a(n > 0) = 2 * A325782(n).

A353854 Length of the trajectory of the composition run-sum transformation (condensation) of the n-th composition in standard order.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 3, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 3, 2, 3, 2, 2, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 2, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 2, 3, 3, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 3, 1, 1, 3, 2, 1, 1, 1, 2, 2, 2, 2

Offset: 0

Gus Wiseman, Jun 01 2022

nonn

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). The run-sum trajectory is obtained by repeatedly taking the run-sum transformation (A353847) until the rank of an anti-run is reached. For example, the trajectory 11 -> 10 -> 8 corresponds to the trajectory (2,1,1) -> (2,2) -> (4), with length a(11) = 3.

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The trajectory of 94685 and the a(94685) = 5 corresponding compositions:
  94685: (2,1,1,4,1,1,2,1,1,2,1)
  86357: (2,2,4,2,2,2,2,1)
  69889: (4,4,8,1)
  65793: (8,8,1)
  65537: (16,1)

Positions of first appearances are A072639.
Positions of 1's are A333489, counted by A003242 (complement A261983).
The version for partitions is A353841.
The last part of the same trajectory is A353855.
This is the rank statistic counted by A353859.
A005811 counts runs in binary expansion.
A011782 counts compositions.
A066099 lists compositions in standard order.
A318928 gives runs-resistance of binary expansion.
A333627 represents the run-lengths of standard compositions.
A353832 represents the run-sum transformation of a partition.
A353840-A353846 pertain to the partition run-sum trajectory.
A353847 represents the run-sum transformation of a composition.
A353853-A353859 pertain to the composition run-sum trajectory.
A353932 lists run-sums of standard compositions, represented by A353847.
Cf. A237685, A238279, A304442, A325277, A333755, A353848, A353849, A353850, A353852, A353860.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[FixedPointList[Total/@Split[#]&,stc[n]]]-1,{n,0,100}]

A353855 Last term of the trajectory of the composition run-sum transformation (condensation) of the n-th composition in standard order.

Original entry on oeis.org

0, 1, 2, 2, 4, 5, 6, 4, 8, 9, 8, 8, 12, 13, 8, 8, 16, 17, 18, 18, 20, 17, 22, 20, 24, 25, 24, 24, 20, 17, 18, 16, 32, 33, 34, 34, 32, 37, 38, 32, 40, 41, 32, 34, 44, 45, 32, 40, 48, 49, 50, 50, 52, 49, 54, 52, 40, 41, 40, 32, 32, 37, 34, 32, 64, 65, 66, 66, 68

Offset: 0

Gus Wiseman, Jun 01 2022

nonn

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). The run-sum trajectory is obtained by repeatedly taking the run-sum transformation (A353847) until the rank of an anti-run is reached. For example, the trajectory 11 -> 10 -> 8, corresponding to (2,1,1) -> (2,2) -> (4), has last term a(11) = 8.

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The trajectory 139 -> 138 -> 136 -> 128 ends with a(139) = 128.

The version for partitions is A353842.
This trajectory has length A353854, firsts A072639, partitions A353841.
A005811 counts runs in binary expansion.
A011782 counts compositions.
A066099 lists compositions in standard order.
A318928 gives runs-resistance of binary expansion.
A325268 counts partitions by omicron, rank statistic A304465.
A333627 ranks the run-lengths of standard compositions.
A351014 counts distinct runs in standard compositions, firsts A351015.
A353840-A353846 pertain to a partition's run-sum trajectory.
A353847 represents a composition's run-sums, partitions A353832.
A353853-A353859 pertain to a composition's run-sum trajectory.
Cf. A237685, A304442, A333381, A333489, A333755, A353848, A353849, A353850, A353852, A353860.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Total[2^Accumulate[Reverse[FixedPoint[Total/@Split[#]&,stc[n]]]]/2],{n,0,100}]

A353858 Number of integer compositions of n with run-sum trajectory ending in a singleton.

Original entry on oeis.org

0, 1, 2, 2, 5, 2, 8, 2, 20, 5, 8, 2, 78, 2, 8, 8, 223, 2, 179, 2, 142, 8, 8, 2, 4808

Offset: 0

Gus Wiseman, Jun 17 2022

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). The run-sum trajectory is obtained by repeatedly taking the run-sums (cf. A353847) until an anti-run composition (A003242) is reached. For example, the composition (2,2,1,1,2) is counted under a(8) because it has the following run-sum trajectory: (2,2,1,1,2) -> (4,2,2) -> (4,4) -> (8).

			The a(0) = 0 through a(8) = 20 compositions:
  .  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
          (11)  (111)  (22)    (11111)  (33)      (1111111)  (44)
                       (112)            (222)                (224)
                       (211)            (1113)               (422)
                       (1111)           (2112)               (1124)
                                        (3111)               (2114)
                                        (11211)              (2222)
                                        (111111)             (4112)
                                                             (4211)
                                                             (11114)
                                                             (21122)
                                                             (22112)
                                                             (41111)
                                                             (111122)
                                                             (112112)
                                                             (211211)
                                                             (221111)
                                                             (1111211)
                                                             (1121111)
                                                             (11111111)

The version for partitions is A353845, ranked by A353844.
The trajectory itself is A353853, last part A353855.
The lengths of trajectories of standard compositions are A353854.
This is column k = 1 of A353856, for partitions A353843.
These compositions are ranked by A353857.
A011782 counts compositions.
A066099 lists compositions in standard order.
A238279 and A333755 count compositions by number of runs.
A275870 counts collapsible partitions, ranked by A300273.
A333489 ranks anti-runs, counted by A003242 (complement A261983).
A353840-A353846 pertain to partition run-sum trajectory.
A353847 represents the run-sums of a composition, partitions A353832.
A353851 counts compositions with equal run-sums, ranked by A353848.
A353859 counts compositions by length of run-sum trajectory.
A353860 counts collapsible compositions.
A353932 lists run-sums of standard compositions.
Cf. A072639, A237685, A304442, A304465, A318928, A353833, A353841, A353849, A353850, A353852.

Table[Length[Select[Join@@Permutations/@ IntegerPartitions[n], Length[FixedPoint[Total/@Split[#]&,#]]==1&]],{n,0,15}]

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

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jun 07 2022

nonn

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).

			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)

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

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.

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

a(n) = {my(t=1); if(n==2, t<<=1, for(k=3, n, t = (t<Andrew Howroyd, Jan 01 2023

Offset corrected and terms a(7) and beyond from Andrew Howroyd, Jan 01 2023

A353842 Last part of the trajectory of the partition run-sum transformation of n, using Heinz numbers.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 5, 7, 10, 11, 7, 13, 14, 15, 7, 17, 14, 19, 15, 21, 22, 23, 15, 13, 26, 13, 21, 29, 30, 31, 11, 33, 34, 35, 21, 37, 38, 39, 13, 41, 42, 43, 33, 35, 46, 47, 21, 19, 26, 51, 39, 53, 26, 55, 35, 57, 58, 59, 35, 61, 62, 19, 13, 65, 66, 67, 51

Offset: 1

Gus Wiseman, May 25 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

The run-sum trajectory is obtained by repeatedly taking the run-sum transformation (A353832) until a squarefree number is reached. For example, the trajectory 12 -> 9 -> 7 corresponds to the partitions (2,1,1) -> (2,2) -> (4).

			The partition run-sum trajectory of 87780 is: 87780 -> 65835 -> 51205 -> 19855 -> 2915, so a(87780) = 2915.

The fixed points and image are A005117.
For run-lengths instead of sums we have A304464/A304465, counted by A325268.
These are the row-ends of A353840.
Other sequences pertaining to partition trajectory are A353841-A353846.
The version for compositions is A353855, run-ends of A353853.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A182850 and A323014 give frequency depth.
A300273 ranks collapsible partitions, counted by A275870.
A353832 represents the operation of taking run-sums of a partition.
A353833 ranks partitions with all equal run-sums, counted by A304442.
A353835 counts distinct run-sums of prime indices, weak A353861.
A353866 ranks rucksack partitions, counted by A353864.
Cf. A071625, A073093, A181819, A325239, A325277, A353834, A353838, A353847, A353865, A353867.

Table[NestWhile[Times@@Prime/@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>PrimePi[p]*k]&,n,!SquareFreeQ[#]&],{n,100}]

A354578 Number of ways to choose a divisor of each part of the n-th composition in standard order such that no adjacent divisors are equal.

Original entry on oeis.org

1, 1, 2, 0, 2, 1, 1, 0, 3, 1, 2, 0, 1, 1, 0, 0, 2, 2, 3, 0, 3, 1, 1, 0, 2, 1, 1, 0, 0, 0, 0, 0, 4, 1, 4, 0, 2, 2, 1, 0, 4, 2, 2, 0, 1, 1, 0, 0, 1, 2, 2, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 3, 0, 5, 2, 2, 0, 5, 1, 3, 0, 1, 1, 0, 0, 3, 3, 5, 0, 3, 1, 1

Offset: 0

Gus Wiseman, Jun 11 2022

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). Then a(n) is the number of integer compositions whose run-sums constitute the n-th composition in standard order (graded reverse-lexicographic, A066099).

			The terms 2^(n - 1) through 2^n - 1 sum to 2^n. As a triangle:
  1
  1
  2 0
  2 1 1 0
  3 1 2 0 1 1 0 0
  2 2 3 0 3 1 1 0 2 1 1 0 0 0 0 0
The a(n) compositions for selected n:
  n=1: n=2:   n=8:       n=32:          n=68:        n=130:
----------------------------------------------------------------------
  (1)  (2)    (4)        (6)            (4,3)        (6,2)
       (1,1)  (2,2)      (3,3)          (2,2,3)      (3,3,2)
              (1,1,1,1)  (2,2,2)        (4,1,1,1)    (6,1,1)
                         (1,1,1,1,1,1)  (1,1,1,1,3)  (3,3,1,1)
                                        (2,2,1,1,1)  (2,2,2,1,1)
                                                     (1,1,1,1,1,1,2)

First column is 1 followed by A000005.
Row-sums are A011782.
Standard compositions are listed by A066099.
Positions of 0's are A354904.
Positions of first appearances are A354905.
A003242 counts anti-run compositions, ranked by A333489.
A005811 counts runs in binary expansion.
A300273 ranks collapsible partitions, counted by A275870.
A353838 ranks partitions with all distinct run-sums, counted by A353837.
A353851 counts compositions with all equal run-sums, ranked by A353848.
A353840-A353846 pertain to partition run-sum trajectory.
A353852 ranks compositions with all distinct run-sums, counted by A353850.
A353853-A353859 pertain to composition run-sum trajectory.
A353860 counts collapsible compositions.
A353863 counts run-sum-complete partitions.
A354584 gives run-sums of prime indices, rows ranked by A353832.
Cf. A029837, A124767, A175413, A238279/A333755, A333381, A353847, A353849.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
antirunQ[y_]:=Length[Split[y]]==Length[y];
Table[Length[Select[Tuples[Divisors/@stc[n]],antirunQ]],{n,0,30}]

A353844 Starting with the multiset of prime indices of n, repeatedly take the multiset of run-sums until you reach a squarefree number. This number is prime (or 1) iff n belongs to the sequence.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 40, 41, 43, 47, 49, 53, 59, 61, 63, 64, 67, 71, 73, 79, 81, 83, 84, 89, 97, 101, 103, 107, 109, 112, 113, 121, 125, 127, 128, 131, 137, 139, 144, 149, 151, 157, 163, 167, 169, 173, 179

Offset: 1

Gus Wiseman, May 26 2022

nonn

The run-sums transformation is described by Kimberling at A237685 and A237750.
The runs of a sequence are its maximal consecutive constant subsequences. For example, the runs of {1,1,1,2,2,3,4} are {1,1,1}, {2,2}, {3}, {4}, with sums {3,3,4,4}.
Note that the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so this sequence lists Heinz numbers of partitions whose run-sum trajectory reaches an empty set or singleton.

			The terms together with their prime indices begin:
      1: {}            25: {3,3}           64: {1,1,1,1,1,1}
      2: {1}           27: {2,2,2}         67: {19}
      3: {2}           29: {10}            71: {20}
      4: {1,1}         31: {11}            73: {21}
      5: {3}           32: {1,1,1,1,1}     79: {22}
      7: {4}           37: {12}            81: {2,2,2,2}
      8: {1,1,1}       40: {1,1,1,3}       83: {23}
      9: {2,2}         41: {13}            84: {1,1,2,4}
     11: {5}           43: {14}            89: {24}
     12: {1,1,2}       47: {15}            97: {25}
     13: {6}           49: {4,4}          101: {26}
     16: {1,1,1,1}     53: {16}           103: {27}
     17: {7}           59: {17}           107: {28}
     19: {8}           61: {18}           109: {29}
     23: {9}           63: {2,2,4}        112: {1,1,1,1,4}
The trajectory 60 -> 45 -> 35 ends in a nonprime number 35, so 60 is not in the sequence.
The trajectory 84 -> 63 -> 49 -> 19 ends in a prime number 19, so 84 is in the sequence.

This sequence is a subset of A300273, counted by A275870.
The version for compositions is A353857, counted by A353847.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A304442 counts partitions with all equal run-sums.
A353851 counts compositions with all equal run-sums, ranked by A353848.
A325268 counts partitions by omicron, rank statistic A304465.
A353832 represents the operation of taking run-sums of a partition.
A353833 ranks partitions with all equal run-sums, nonprime A353834.
A353835 counts distinct run-sums of prime indices, weak A353861.
A353838 ranks partitions with all distinct run-sums, counted by A353837.
A353840-A353846 pertain to partition run-sum trajectory.
A353853-A353859 pertain to composition run-sum trajectory.
A353866 ranks rucksack partitions, counted by A353864.
Cf. A005811, A073093, A130091, A181819, A182857, A304660, A325239, A325277, A353839, A353862, A353867.

ope[n_]:=Times@@Prime/@Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>PrimePi[p]*k];
Select[Range[100],#==1||PrimeQ[NestWhile[ope,#,!SquareFreeQ[#]&]]&]

A353845 Number of integer partitions of n such that if you repeatedly take the multiset of run-sums (or condensation), you eventually reach an empty set or singleton.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 5, 2, 8, 3, 5, 2, 15, 2, 5, 4, 18, 2, 13, 2, 14, 4, 5, 2, 62, 3, 5, 5, 14, 2, 18, 2, 48, 4, 5, 4, 71, 2, 5, 4, 54, 2, 18, 2, 14, 10, 5, 2, 374, 3, 9, 4, 14, 2, 37, 4, 54, 4, 5, 2, 131

Offset: 0

Gus Wiseman, May 26 2022

nonn

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).

			The a(1) = 1 through a(8) = 8 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (11111)  (33)      (1111111)  (44)
                    (211)            (222)                (422)
                    (1111)           (3111)               (2222)
                                     (111111)             (4211)
                                                          (41111)
                                                          (221111)
                                                          (11111111)
For example, the partition (3,2,2,2,1,1,1) has trajectory: (1,1,1,2,2,2,3) -> (3,3,6) -> (6,6) -> (12), so is counted under a(12).

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

Dominated by A018818 (partitions into divisors).
The version for compositions is A353858.
A275870 counts collapsible partitions, ranked by A300273.
A304442 counts partitions with all equal run-sums, ranked by A353833.
A325268 counts partitions by omicron, rank statistic A304465.
A353832 represents the operation of taking run-sums of a partition.
A353837 counts partitions with all distinct run-sums, ranked by A353838.
A353840-A353846 pertain to partition run-sum trajectory.
A353847-A353859 pertain to composition run-sum trajectory.
A353864 counts rucksack partitions, ranked by A353866.
Cf. A000041, A008284, A181819, A225485, A325239, A325277, A325280, A326370, A353834, A353839, A353865.

Table[Length[Select[IntegerPartitions[n], Length[NestWhile[Sort[Total/@Split[#]]&,#,!UnsameQ@@#&]]<=1&]],{n,0,30}]

cp's OEIS Frontend

A353862 Greatest run-sum of the prime indices of n.

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A353743 Least number with run-sum trajectory of length k; a(0) = 1.

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Formula

A353854 Length of the trajectory of the composition run-sum transformation (condensation) of the n-th composition in standard order.

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A353855 Last term of the trajectory of the composition run-sum transformation (condensation) of the n-th composition in standard order.

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A353858 Number of integer compositions of n with run-sum trajectory ending in a singleton.

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

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A353842 Last part of the trajectory of the partition run-sum transformation of n, using Heinz numbers.

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A354578 Number of ways to choose a divisor of each part of the n-th composition in standard order such that no adjacent divisors are equal.

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A353844 Starting with the multiset of prime indices of n, repeatedly take the multiset of run-sums until you reach a squarefree number. This number is prime (or 1) iff n belongs to the sequence.

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