A333755 -id:A333755 - cp's OEIS frontend

A353847 Composition run-sum transformation in terms of standard composition numbers. The a(k)-th composition in standard order is the sequence of run-sums of the k-th composition in standard order. Takes each index of a row of A066099 to the index of the row consisting of its run-sums.

0, 1, 2, 2, 4, 5, 6, 4, 8, 9, 8, 10, 12, 13, 10, 8, 16, 17, 18, 18, 20, 17, 22, 20, 24, 25, 24, 26, 20, 21, 18, 16, 32, 33, 34, 34, 32, 37, 38, 36, 40, 41, 32, 34, 44, 45, 42, 40, 48, 49, 50, 50, 52, 49, 54, 52, 40, 41, 40, 42, 36, 37, 34, 32, 64, 65, 66, 66

Offset: 0

Gus Wiseman, May 30 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 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.

			As a triangle:
   0
   1
   2  2
   4  5  6  4
   8  9  8 10 12 13 10  8
  16 17 18 18 20 17 22 20 24 25 24 26 20 21 18 16
These are the standard composition numbers of the following compositions (transposed):
  ()  (1)  (2)  (3)    (4)      (5)
           (2)  (2,1)  (3,1)    (4,1)
                (1,2)  (4)      (3,2)
                (3)    (2,2)    (3,2)
                       (1,3)    (2,3)
                       (1,2,1)  (4,1)
                       (2,2)    (2,1,2)
                       (4)      (2,3)
                                (1,4)
                                (1,3,1)
                                (1,4)
                                (1,2,2)
                                (2,3)
                                (2,2,1)
                                (3,2)
                                (5)

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

Standard compositions are listed by A066099.
The version for partitions is A353832.
The run-sums themselves are listed by A353932, with A353849 distinct terms.
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.
Cf. A003242. A175413, A181819, A238279, A274174, A333381, A333489, A333755, A353835, A353839, A353864.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
Table[stcinv[Total/@Split[stc[n]]],{n,0,100}]

A353833 Numbers whose multiset of prime indices has all equal run-sums.

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, 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 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.

The sequence of runs of a sequence consists of its maximal consecutive constant subsequences when read left-to-right. 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 prime indices of 12 are {1,1,2}, with run-sums (2,2), so 12 is in the sequence.

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

For parts instead of run-sums we have A000961, counted by A000005.
For run-lengths instead of run-sums we have A072774, counted by A047966.
These partitions are counted by A304442.
These are the positions of powers of primes in A353832.
The restriction to nonprimes is A353834.
For distinct instead of equal run-sums we have A353838, counted by A353837.
The version for compositions is A353848, counted by A353851.
A001222 counts prime factors, distinct A001221.
A005811 counts runs in binary expansion, distinct run-lengths A165413.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A300273 ranks collapsible partitions, counted by A275870.
A353835 counts distinct run-sums of prime indices, weak A353861.
A353840-A353846 deal with iterated run-sums for partitions.
A353862 gives greatest run-sum of prime indices, least A353931.
A353866 ranks rucksack partitions, counted by A353864.
Cf. A007947, A071625, A073093, A181819, A238279, A304660, A323014, A333755, A353839.

Select[Range[100],SameQ@@Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]*k]&]

A261982 Number of compositions of n with some part repeated.

Original entry on oeis.org

0, 0, 1, 1, 5, 11, 21, 51, 109, 229, 455, 959, 1947, 3963, 7999, 16033, 32333, 64919, 130221, 260967, 522733, 1045825, 2093855, 4189547, 8382315, 16768455, 33543127, 67093261, 134193413, 268404995, 536829045, 1073686083, 2147408773, 4294869253, 8589803783

Offset: 0

Alois P. Heinz, Sep 07 2015

nonn

Also compositions matching the pattern (1,1). - Gus Wiseman, Jun 23 2020

			a(2) = 1: 11.
a(3) = 1: 111.
a(4) = 5: 22, 211, 121, 112, 1111.

Alois P. Heinz, Table of n, a(n) for n = 0..3322
Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Row sums of A261981 and of A262191.
Cf. A262047.
The version for patterns is A019472.
The (1,1)-avoiding version is A032020.
The case of partitions is A047967.
(1,1,1)-matching compositions are counted by A335455.
Patterns matched by compositions are counted by A335456.
(1,1)-matching compositions are ranked by A335488.
Cf. A011782, A056986, A106356, A181796, A238279, A269134, A333755.

b:= proc(n, k) option remember; `if`(k<0 or n<0, 0,
      `if`(k=0, `if`(n=0, 1, 0), b(n-k, k) +k*b(n-k, k-1)))
    end:
a:= n-> ceil(2^(n-1))-add(b(n, k), k=0..floor((sqrt(8*n+1)-1)/2)):
seq(a(n), n=0..40);

b[n_, k_] := b[n, k] = If[k<0 || n<0, 0, If[k==0, If[n==0, 1, 0], b[n-k, k] + k*b[n-k, k-1]]]; a[n_] := Ceiling[2^(n-1)]-Sum[b[n, k], {k, 0, Floor[ (Sqrt[8n+1]-1)/2]}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 08 2017, translated from Maple *)
Table[Length[Join@@Permutations/@Select[IntegerPartitions[n],Length[#]>Length[Split[#]]&]],{n,0,10}] (* Gus Wiseman, Jun 24 2020 *)

a(n) = A011782(n) - A032020(n).

G.f.: (1 - x) / (1 - 2*x) - Sum_{k>=0} k! * x^(k*(k + 1)/2) / Product_{j=1..k} (1 - x^j). - Ilya Gutkovskiy, Jan 30 2020

A373948 Run-compression encoded as a transformation of compositions in standard order.

Original entry on oeis.org

0, 1, 2, 1, 4, 5, 6, 1, 8, 9, 2, 5, 12, 13, 6, 1, 16, 17, 18, 9, 20, 5, 22, 5, 24, 25, 6, 13, 12, 13, 6, 1, 32, 33, 34, 17, 4, 37, 38, 9, 40, 41, 2, 5, 44, 45, 22, 5, 48, 49, 50, 25, 52, 13, 54, 13, 24, 25, 6, 13, 12, 13, 6, 1, 64, 65, 66, 33, 68, 69, 70, 17, 72

Offset: 0

Gus Wiseman, Jun 24 2024

nonn

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.
We define the (run-) compression of a sequence to be the anti-run obtained by reducing each run of repeated parts to a single part. Alternatively, compression removes all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has compression (1,2,1).
For the present sequence, the a(n)-th composition in standard order is obtained by compressing the n-th composition in standard order.

			The standard compositions and their compressions begin:
   0: ()        -->  0: ()
   1: (1)       -->  1: (1)
   2: (2)       -->  2: (2)
   3: (1,1)     -->  1: (1)
   4: (3)       -->  4: (3)
   5: (2,1)     -->  5: (2,1)
   6: (1,2)     -->  6: (1,2)
   7: (1,1,1)   -->  1: (1)
   8: (4)       -->  8: (4)
   9: (3,1)     -->  9: (3,1)
  10: (2,2)     -->  2: (2)
  11: (2,1,1)   -->  5: (2,1)
  12: (1,3)     --> 12: (1,3)
  13: (1,2,1)   --> 13: (1,2,1)
  14: (1,1,2)   -->  6: (1,2)
  15: (1,1,1,1) -->  1: (1)

Gus Wiseman, Statistics, classes, and transformations of standard compositions

Positions of 1's are A000225.
The image is A333489, counted by A003242.
Sum of standard composition for a(n) is given by A373953, length A124767.
A037201 gives compression of first differences of primes, halved A373947.
A066099 lists the parts of all compositions in standard order.
A114901 counts compositions with no isolated parts.
A116861 counts partitions by compressed sum, by length A116608.
A240085 counts compositions with no unique parts.
A333755 counts compositions by compressed length.
A373949 counts compositions by compressed sum, opposite A373951.
Cf. A106356, A124762, A238130, A238279, A272919, A333381, A333382, A333627, A373952, A373954.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
Table[stcinv[First/@Split[stc[n]]],{n,0,30}]

A029837(a(n)) = A373953(n).

A000120(a(n)) = A124767(n).

A344604 Number of alternating compositions of n, including twins (x,x).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 13, 19, 30, 48, 76, 118, 187, 293, 461, 725, 1140, 1789, 2815, 4422, 6950, 10924, 17169, 26979, 42405, 66644, 104738, 164610, 258708, 406588, 639010, 1004287, 1578364, 2480606, 3898600, 6127152, 9629624, 15134213, 23785389, 37381849, 58750469

Offset: 0

Gus Wiseman, May 27 2021

nonn

We define a composition to be alternating including twins (x,x) if there are no adjacent triples (..., x, y, z, ...) where x <= y <= z or x >= y >= z. Except in the case of twins (x,x), all such compositions are anti-runs (A003242). These compositions avoid the weak consecutive patterns (1,2,3) and (3,2,1), the strict version being A344614.

The version without twins (x,x) is A025047 (alternating compositions).

			The a(1) = 1 through a(7) = 19 compositions:
  (1)  (2)   (3)   (4)    (5)    (6)     (7)
       (11)  (12)  (13)   (14)   (15)    (16)
             (21)  (22)   (23)   (24)    (25)
                   (31)   (32)   (33)    (34)
                   (121)  (41)   (42)    (43)
                          (131)  (51)    (52)
                          (212)  (132)   (61)
                                 (141)   (142)
                                 (213)   (151)
                                 (231)   (214)
                                 (312)   (232)
                                 (1212)  (241)
                                 (2121)  (313)
                                         (412)
                                         (1213)
                                         (1312)
                                         (2131)
                                         (3121)
                                         (12121)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

A001250 counts alternating permutations.
A005649 counts anti-run patterns.
A025047 counts alternating or wiggly compositions, also A025048, A025049.
A106356 counts compositions by number of maximal anti-runs.
A114901 counts compositions where each part is adjacent to an equal part.
A325534 counts separable partitions.
A325535 counts inseparable partitions.
A344605 counts alternating patterns including twins.
A344606 counts alternating permutations of prime factors including twins.
Counting compositions by patterns:
- A011782 no conditions.
- A003242 avoiding (1,1) adjacent.
- A102726 avoiding (1,2,3).
- A106351 avoiding (1,1) adjacent by sum and length.
- A128695 avoiding (1,1,1) adjacent.
- A128761 avoiding (1,2,3) adjacent.
- A232432 avoiding (1,1,1).
- A335456 all patterns.
- A335457 all patterns adjacent.
- A335514 matching (1,2,3).
- A344614 avoiding (1,2,3) and (3,2,1) adjacent.
- A344615 weakly avoiding (1,2,3) adjacent.
Cf. A000041, A006330, A008965, A238279, A239830, A333213, A238279/A333755, A344612, A344616, A344617, A344618.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MatchQ[#,{_,x_,y_,z_,_}/;x<=y<=z||x>=y>=z]&]],{n,0,15}]

a(n > 0) = A025047(n) + 1 if n is even, otherwise A025047(n). - Gus Wiseman, Nov 03 2021

a(21)-a(40) from Alois P. Heinz, Nov 04 2021

A344614 Number of compositions of n with no adjacent triples (..., x, y, z, ...) where x < y < z or x > y > z.

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 30, 58, 110, 209, 397, 753, 1429, 2711, 5143, 9757, 18511, 35117, 66621, 126389, 239781, 454897, 863010, 1637260, 3106138, 5892821, 11179603, 21209446, 40237641, 76337091, 144823431, 274752731, 521249018, 988891100, 1876081530, 3559220898, 6752400377

Offset: 0

Gus Wiseman, May 27 2021

nonn

These compositions avoid the strict consecutive patterns (1,2,3) and (3,2,1), the weak version being A344604.

			The a(6) = 30 compositions are:
  (6)  (15)  (114)  (1113)  (11112)  (111111)
       (24)  (132)  (1122)  (11121)
       (33)  (141)  (1131)  (11211)
       (42)  (213)  (1212)  (12111)
       (51)  (222)  (1221)  (21111)
             (231)  (1311)
             (312)  (2112)
             (411)  (2121)
                    (2211)
                    (3111)
Missing are: (123), (321).

A001250 counts alternating permutations.
A005649 counts anti-run patterns.
A025047 counts wiggly compositions (ascend: A025048, descend: A025049).
A106356 counts compositions by number of maximal anti-runs.
A114901 counts compositions where each part is adjacent to an equal part.
A325534 counts separable partitions.
A325535 counts inseparable partitions.
A344604 counts wiggly compositions with twins.
A344605 counts wiggly patterns with twins.
A344606 counts wiggly permutations of prime factors with twins.
Counting compositions by patterns:
- A003242 avoiding (1,1) adjacent.
- A011782 no conditions.
- A106351 avoiding (1,1) adjacent by sum and length.
- A128695 avoiding (1,1,1) adjacent.
- A128761 avoiding (1,2,3).
- A232432 avoiding (1,1,1).
- A335456 all patterns.
- A335457 all patterns adjacent.
- A335514 matching (1,2,3).
- A344604 weakly avoiding (1,2,3) and (3,2,1) adjacent.
- A344614 avoiding (1,2,3) and (3,2,1) adjacent.
- A344615 weakly avoiding (1,2,3) adjacent.
Cf. A000041, A006330, A008965, A049774, A056986, A238279/A333755, A333213, A335515, A344612, A344652.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MatchQ[#,{_,x_,y_,z_,_}/;xy>z]&]],{n,0,15}]

More terms from Bert Dobbelaere, Jun 12 2021

A353850 Number of integer compositions of n with all distinct run-sums.

Original entry on oeis.org

1, 1, 2, 4, 5, 12, 24, 38, 52, 111, 218, 286, 520, 792, 1358, 2628, 4155, 5508, 9246, 13182, 23480, 45150, 54540, 94986, 146016, 213725, 301104, 478586, 851506, 1302234, 1775482, 2696942, 3746894, 6077784, 8194466, 12638334, 21763463, 28423976, 45309850, 62955524, 94345474

Offset: 0

Gus Wiseman, May 31 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(0) = 1 through a(5) = 12 compositions:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (13)    (14)
                 (21)   (22)    (23)
                 (111)  (31)    (32)
                        (1111)  (41)
                                (113)
                                (122)
                                (221)
                                (311)
                                (1112)
                                (2111)
                                (11111)
For n=4, (211) is invalid because the two runs (2) and (11) have the same sum. - _Joseph Likar_, Aug 04 2023

Joseph Likar, Table of n, a(n) for n = 0..120

For distinct parts instead of run-sums we have A032020.
For distinct multiplicities instead of run-sums we have A242882.
For distinct run-lengths instead of run-sums we have A329739, ptns A098859.
For runs instead of run-sums we have A351013.
For partitions we have A353837, ranked by A353838 (complement A353839).
For equal instead of distinct run-sums we have A353851, ptns A304442.
These compositions are ranked by A353852.
The weak version (rucksack compositions) is A354580, ranked by A354581.
A003242 counts anti-run compositions, ranked by A333489.
A005811 counts runs in binary expansion.
A011782 counts compositions.
A175413 lists numbers whose binary expansion has all distinct runs.
A351014 counts distinct runs in standard compositions, firsts A351015.
A353847 gives composition run-sum transformation.
A353929 counts distinct runs in binary expansion, firsts A353930.
Cf. A238279, A333755, A351016, A351017, A353832, A353848, A353849, A353853-A353859, A353860, A353863, A353932.

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

Terms a(21) and onwards from Joseph Likar, Aug 04 2023

A373953 Sum of run-compression of the n-th integer composition in standard order.

Original entry on oeis.org

0, 1, 2, 1, 3, 3, 3, 1, 4, 4, 2, 3, 4, 4, 3, 1, 5, 5, 5, 4, 5, 3, 5, 3, 5, 5, 3, 4, 4, 4, 3, 1, 6, 6, 6, 5, 3, 6, 6, 4, 6, 6, 2, 3, 6, 6, 5, 3, 6, 6, 6, 5, 6, 4, 6, 4, 5, 5, 3, 4, 4, 4, 3, 1, 7, 7, 7, 6, 7, 7, 7, 5, 7, 4, 5, 6, 7, 7, 6, 4, 7, 7, 7, 6, 5, 3, 5

Offset: 0

Gus Wiseman, Jun 25 2024

nonn

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.

We define the (run-) compression of a sequence to be the anti-run obtained by reducing each run of repeated parts to a single part. Alternatively, compression removes all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has compression (1,2,1).

			The standard compositions and their compressions and compression sums begin:
   0: ()        --> ()      --> 0
   1: (1)       --> (1)     --> 1
   2: (2)       --> (2)     --> 2
   3: (1,1)     --> (1)     --> 1
   4: (3)       --> (3)     --> 3
   5: (2,1)     --> (2,1)   --> 3
   6: (1,2)     --> (1,2)   --> 3
   7: (1,1,1)   --> (1)     --> 1
   8: (4)       --> (4)     --> 4
   9: (3,1)     --> (3,1)   --> 4
  10: (2,2)     --> (2)     --> 2
  11: (2,1,1)   --> (2,1)   --> 3
  12: (1,3)     --> (1,3)   --> 4
  13: (1,2,1)   --> (1,2,1) --> 4
  14: (1,1,2)   --> (1,2)   --> 3
  15: (1,1,1,1) --> (1)     --> 1

Gus Wiseman, Statistics, classes, and transformations of standard compositions

Positions of 1's are A000225.
Counting partitions by this statistic gives A116861, by length A116608.
For length instead of sum we have A124767, counted by A238279 and A333755.
Compositions counted by this statistic are A373949, opposite A373951.
A037201 gives compression of first differences of primes, halved A373947.
A066099 lists the parts of all compositions in standard order.
A114901 counts compositions with no isolated parts.
A240085 counts compositions with no unique parts.
A333489 ranks anti-runs, counted by A003242.
Cf. A106356, A124762, A238130, A238343, A272919, A285981, A333381, A333382, A333627, A373952, A373954.

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

a(n) = A029837(A373948(n)).

A353848 Numbers k such that the k-th composition in standard order (row k of A066099) has all equal run-sums.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 8, 10, 11, 14, 15, 16, 31, 32, 36, 39, 42, 46, 59, 60, 63, 64, 127, 128, 136, 138, 143, 168, 170, 175, 187, 238, 248, 250, 255, 256, 292, 316, 487, 511, 512, 528, 543, 682, 750, 955, 1008, 1023, 1024, 2047, 2048, 2080, 2084, 2090, 2111, 2184

Offset: 0

Gus Wiseman, May 30 2022

nonn

Every sequence can be uniquely split into non-overlapping runs, read left-to-right. 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 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 terms together with their binary expansions and corresponding compositions begin:
     0:       0  ()
     1:       1  (1)
     2:      10  (2)
     3:      11  (1,1)
     4:     100  (3)
     7:     111  (1,1,1)
     8:    1000  (4)
    10:    1010  (2,2)
    11:    1011  (2,1,1)
    14:    1110  (1,1,2)
    15:    1111  (1,1,1,1)
    16:   10000  (5)
    31:   11111  (1,1,1,1,1)
    32:  100000  (6)
    36:  100100  (3,3)
    39:  100111  (3,1,1,1)
    42:  101010  (2,2,2)
    46:  101110  (2,1,1,2)
    59:  111011  (1,1,2,1,1)
    60:  111100  (1,1,1,3)
For example:
- The 59th composition in standard order is (1,1,2,1,1), with run-sums (2,2,2), so 59 is in the sequence.
- The 2298th composition in standard order is (4,1,1,1,1,2,2), with run-sums (4,4,4), so 2298 is in the sequence.
- The 2346th composition in standard order is (3,3,2,2,2), with run-sums (6,6), so 2346 is in the sequence.

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

Standard compositions are listed by A066099.
For equal lengths instead of sums we have A353744, counted by A329738.
The version for partitions is A353833, counted by A304442.
These compositions are counted by A353851.
The distinct instead of equal version is A353852, counted by A353850.
The run-sums themselves are listed by A353932, with A353849 distinct terms.
A005811 counts runs in binary expansion.
A300273 ranks collapsible partitions, counted by A275870.
A351014 counts distinct runs in standard compositions, firsts A351015.
A353840-A353846 pertain to partition run-sum trajectory.
A353847 represents the run-sum transformation for compositions.
A353853-A353859 pertain to composition run-sum trajectory.
A353860 counts collapsible compositions.
A353863 counts run-sum-complete partitions.
Cf. A003242, A047966, A106356, A140690, A238279, A274174, A333381, A333489, A333755, A353832, A353864.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],SameQ@@Total/@Split[stc[#]]&]

A353849(a(n)) = 1.

A037201 Differences between consecutive primes (A001223) but with repeats omitted.

Original entry on oeis.org

1, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 4, 6, 2, 10, 2, 4, 2, 12, 4, 2, 4, 6, 2, 10, 6, 2, 6, 4, 2, 10, 14, 4, 2, 4, 14, 6, 10, 2, 4, 6, 8, 6, 4, 6, 8, 4, 8, 10, 2, 10, 2, 6, 4, 6, 8, 4

Offset: 1

Naohiro Nomoto

Also the run-compression of the sequence of first differences of prime numbers, where we define the run-compression of a sequence to be the anti-run obtained by reducing each run of repeated parts to a single part. Alternatively, we can remove all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has run-compression (1,2,1). - Gus Wiseman, Sep 16 2024

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for primes, gaps between

This is the run-compression of A001223 = first differences of A000040.
The repeats were at positions A064113 before being omitted.
Adding up runs instead of compressing them gives A373822.
The even terms halved are A373947.
For prime-powers instead of prime numbers we have A376308.
Positions of first appearances are A376520, sorted A376521.
A003242 counts compressed compositions.
A333254 lists run-lengths of differences between consecutive primes.
A373948 encodes compression using compositions in standard order.
Cf. A007921, A030173, A053289, A106356, A114901, A116608, A116861, A124767, A238130, A333755, A335406, A373954.

a037201 n = a037201_list !! (n-1)
a037201_list = f a001223_list where
   f (x:xs@(x':_)) | x == x'   = f xs
                   | otherwise = x : f xs
-- Reinhard Zumkeller, Feb 27 2012

Flatten[Split[Differences[Prime[Range[150]]]]/.{(k_)..}:>k] (* based on a program by Harvey P. Dale, Jun 21 2012 *)

t=0;p=2;forprime(q=3,1e3,if(q-p!=t,print1(q-p", "));t=q-p;p=q) \\ Charles R Greathouse IV, Feb 27 2012

a(n>1) = 2*A373947(n-1). - Gus Wiseman, Sep 16 2024

Offset corrected by Reinhard Zumkeller, Feb 27 2012

cp's OEIS Frontend

A353847 Composition run-sum transformation in terms of standard composition numbers. The a(k)-th composition in standard order is the sequence of run-sums of the k-th composition in standard order. Takes each index of a row of A066099 to the index of the row consisting of its run-sums.

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A353833 Numbers whose multiset of prime indices has all equal run-sums.

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A261982 Number of compositions of n with some part repeated.

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A373948 Run-compression encoded as a transformation of compositions in standard order.

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A344604 Number of alternating compositions of n, including twins (x,x).

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A344614 Number of compositions of n with no adjacent triples (..., x, y, z, ...) where x < y < z or x > y > z.

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A353850 Number of integer compositions of n with all distinct run-sums.

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A373953 Sum of run-compression of the n-th integer composition in standard order.

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