A124768 -id:A124768 - cp's OEIS frontend

A333230 Positions of weak ascents in the sequence of differences between primes.

1, 2, 3, 5, 7, 8, 10, 13, 14, 15, 17, 20, 22, 23, 26, 28, 29, 31, 33, 35, 36, 38, 39, 41, 43, 45, 46, 49, 50, 52, 54, 55, 57, 60, 61, 64, 65, 67, 69, 70, 71, 73, 75, 76, 78, 79, 81, 83, 85, 86, 89, 90, 93, 95, 96, 98, 100, 102, 104, 105, 107, 109, 110, 113

Offset: 1

Gus Wiseman, Mar 18 2020

Partial sums of A333252.

			The prime gaps split into the following strictly decreasing subsequences: (1), (2), (2), (4,2), (4,2), (4), (6,2), (6,4,2), (4), (6), (6,2), (6,4,2), (6,4), (6), (8,4,2), ...

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Thomas Bloom, Let d_n = p_(n+1)-p_n. The set of n such that d_(n+1) >= d_n has density 1/2, and similarly for d_(n+1) <= d_n. Furthermore, there are infinitely many n such that d_(n+1) = d_n, Erdős Problems.
Terence Tao, Erdős problem database, see no. 218.

The version for the Kolakoski sequence is A022297.
The version for equal differences is A064113.
The version for strict ascents is A258025.
The version for strict descents is A258026.
The version for distinct differences is A333214.
The version for weak descents is A333231.
First differences are A333252 (if the first term is 0).
Prime gaps are A001223.
Weakly decreasing runs of standard compositions are counted by A124765.
Weakly increasing runs of standard compositions are counted by A124766.
Strictly increasing runs of standard compositions are counted by A124768.
Strictly decreasing runs of standard compositions are counted by A124769.
Runs of prime gaps with nonzero differences are A333216.
Cf. A000040, A000720, A036263, A054819, A084758, A333212, A333213, A333215.

Accumulate[Length/@Split[Differences[Array[Prime,100]],#1>#2&]]//Most
(* or *)
Select[Range[100],Prime[#+1]-Prime[#]<=Prime[#+2]-Prime[#+1]&]

Numbers k such that prime(k+2) - 2*prime(k+1) + prime(k) >= 0.

A374744 Numbers k such that the leaders of weakly decreasing runs in the k-th composition in standard order (A066099) are identical.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 15, 16, 17, 18, 19, 21, 22, 23, 31, 32, 33, 34, 35, 36, 37, 39, 42, 43, 45, 46, 47, 63, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 76, 79, 85, 86, 87, 90, 91, 93, 94, 95, 127, 128, 129, 130, 131, 132, 133, 135, 136, 137, 138

Offset: 1

Gus Wiseman, Jul 24 2024

nonn

The leaders of weakly decreasing runs in a sequence are obtained by splitting into maximal weakly decreasing subsequences and taking the first term of each.

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 the corresponding compositions begin:
   0: ()
   1: (1)
   2: (2)
   3: (1,1)
   4: (3)
   5: (2,1)
   7: (1,1,1)
   8: (4)
   9: (3,1)
  10: (2,2)
  11: (2,1,1)
  15: (1,1,1,1)
  16: (5)
  17: (4,1)
  18: (3,2)
  19: (3,1,1)
  21: (2,2,1)
  22: (2,1,2)
  23: (2,1,1,1)
  31: (1,1,1,1,1)

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Other types of runs and their counts: A272919 (A000005), A374519 (A374517), A374685 (A374686), A374759 (A374760).
The opposite is A374633, counted by A374631.
For distinct (instead of identical) leaders we have A374701, count A374743.
Positions of constant rows in A374740, opposite A374629, cf. A374630.
Compositions of this type are counted by A374742.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A374748 counts compositions by sum of leaders of weakly decreasing runs.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1) (or sometimes A070939).
- Parts are listed by A066099.
- Adjacent equal pairs are counted by A124762, unequal A333382.
- Number of max runs: A124765, A124766, A124767, A124768, A124769, A333381.
- Ranks of anti-run compositions are A333489, counted by A003242.
- Run-length transform is A333627.
- Run-compression transform is A373948, sum A373953, excess A373954.
- Ranks of contiguous compositions are A374249, counted by A274174.
Cf. A065120, A106356, A238343, A333175, A333213, A335456, A373949, A374635, A374634, A374768.

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

A333215 Lengths of maximal weakly increasing subsequences in the sequence of prime gaps (A001223).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 14 2020

nonn

Prime gaps are differences between adjacent prime numbers.

			The prime gaps split into the following weakly increasing subsequences: (1,2,2,4), (2,4), (2,4,6), (2,6), (4), (2,4,6,6), (2,6), (4), (2,6), (4,6,8), (4), (2,4), (2,4,14), ...

Prime gaps are A001223.
Ones correspond to strong prime quartets A054804.
Weakly increasing runs of compositions in standard order are A124766.
First differences of A258026 (with zero prepended).
The version for the Kolakoski sequence is A332875.
The weakly decreasing version is A333212.
The unequal version is A333216.
Positions of weak ascents in prime gaps are A333230.
The strictly decreasing version is A333252.
The strictly increasing version is A333253.
The equal version is A333254.
Cf. A000040, A000720, A036263, A054819, A064113, A084758, A124765, A124768, A258025, A333213, A333214.

Length/@Split[Differences[Array[Prime,100]],#1<=#2&]//Most

Ones correspond to strong prime quartets (A054804), so the sum of terms up to but not including the n-th one is A000720(A054804(n - 1)).

A375124 Weakly decreasing run-leader transformation for standard compositions.

Original entry on oeis.org

0, 1, 2, 1, 4, 2, 6, 1, 8, 4, 2, 2, 12, 6, 6, 1, 16, 8, 4, 4, 20, 2, 10, 2, 24, 12, 6, 6, 12, 6, 6, 1, 32, 16, 8, 8, 4, 4, 18, 4, 40, 20, 2, 2, 20, 10, 10, 2, 48, 24, 12, 12, 52, 6, 26, 6, 24, 12, 6, 6, 12, 6, 6, 1, 64, 32, 16, 16, 8, 8, 34, 8, 72, 4, 4, 4, 36

Offset: 0

Gus Wiseman, Aug 02 2024

nonn

The a(n)-th composition in standard order lists the leaders of weakly decreasing runs in the n-th composition in standard order.
The leaders of weakly decreasing runs in a sequence are obtained by splitting it into maximal weakly decreasing subsequences and taking the first term of each.
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 813th composition in standard order is (1,3,2,1,2,1), with weakly decreasing runs ((1),(3,2,1),(2,1)), with leaders (1,3,2). This is the 50th composition in standard order, so a(813) = 50.

Gus Wiseman, Statistics, classes, and transformations of standard compositions.
Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Positions of elements of A233564 are A374701, counted by A374743.
Positions of elements of A272919 are A374744, counted by A374742.
Ranks of rows of A374740.
The opposite version is A375123.
The strict version is A375126.
The strict opposite version is A375125.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1) = A070939(n).
- Parts are listed by A066099.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Run-length transform is A333627, sum A070939.
- Run-compression transform is A373948, sum A373953, excess A373954.
- Ranks of contiguous compositions are A374249, counted by A274174.
- Run-sum transformation is A353847.
Six types of runs:
- Count: A124766, A124765, A124768, A124769, A333381, A124767.
- Leaders: A374629, A374740, A374683, A374757, A374515, A374251.
- Rank: A375123, A375124, A375125, A375126, A375127, A373948.
Cf. A065120, A106356, A189076, A238343, A333213, A373949, A374635, A374687, A374748, A374758.

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],GreaterEqual]],{n,0,100}]

A000120(a(n)) = A124765(n).
A065120(a(n)) = A065120(n).
A070939(a(n)) = A374741(n).

A375125 Strictly increasing run-leader transformation for standard compositions.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 1, 7, 8, 9, 10, 11, 1, 3, 3, 15, 16, 17, 18, 19, 2, 21, 5, 23, 1, 3, 6, 7, 3, 7, 7, 31, 32, 33, 34, 35, 36, 37, 9, 39, 2, 5, 42, 43, 5, 11, 11, 47, 1, 3, 6, 7, 1, 13, 3, 15, 3, 7, 14, 15, 7, 15, 15, 63, 64, 65, 66, 67, 68, 69, 17, 71, 4, 73

Offset: 0

Gus Wiseman, Aug 02 2024

nonn

The a(n)-th composition in standard order lists the leaders of strictly increasing runs in the n-th composition in standard order.
The leaders of strictly increasing runs in a sequence are obtained by splitting it into maximal strictly increasing subsequences and taking the first term of each.
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 813th composition in standard order is (1,3,2,1,2,1), with strictly increasing runs ((1,3),(2),(1,2),(1)), with leaders (1,2,1,1). This is the 27th composition in standard order, so a(813) = 27.

Gus Wiseman, Statistics, classes, and transformations of standard compositions.
Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Positions of elements of A233564 are A374698, counted by A374687.
Positions of elements of A272919 are A374685, counted by A374686.
Ranks of rows of A374683.
The weak version is A375123.
The weak opposite version is A375124.
The opposite version is A375126.
Other transformations: A375127, A373948.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1) = A070939(n).
- Parts are listed by A066099.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Run-length transform is A333627.
- Run-compression transform is A373948, sum A373953, excess A373954.
- Ranks of contiguous compositions are A374249, counted by A274174.
- Run-sum transformation is A353847.
Six types of runs:
- Count: A124766, A124765, A124768, A124769, A333381, A124767.
- Leaders: A374629, A374740, A374683, A374757, A374515, A374251.
- Rank: A375123, A375124, A375125, A375126, A375127, A373948.
Cf. A065120, A106356, A189076, A238343, A333213, A373949, A374634, A374635, A374684, A374700, A375128.

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],Less]],{n,0,100}]

A000120(a(n)) = A124768(n).
A065120(a(n)) = A065120(n).
A070939(a(n)) = A374684(n).

A375126 Strictly decreasing run-leader transformation for standard compositions.

Original entry on oeis.org

0, 1, 2, 3, 4, 2, 6, 7, 8, 4, 10, 5, 12, 6, 14, 15, 16, 8, 4, 9, 20, 10, 10, 11, 24, 12, 26, 13, 28, 14, 30, 31, 32, 16, 8, 17, 36, 4, 18, 19, 40, 20, 42, 21, 20, 10, 22, 23, 48, 24, 12, 25, 52, 26, 26, 27, 56, 28, 58, 29, 60, 30, 62, 63, 64, 32, 16, 33, 8, 8

Offset: 0

Gus Wiseman, Aug 02 2024

nonn

The a(n)-th composition in standard order lists the leaders of strictly decreasing runs in the n-th composition in standard order.
The leaders of strictly decreasing runs in a sequence are obtained by splitting it into maximal strictly decreasing subsequences and taking the first term of each.
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.
Does this sequence contain all nonnegative integers?

			The 813th composition in standard order is (1,3,2,1,2,1), with strictly decreasing runs ((1),(3,2,1),(2,1)), with leaders (1,3,2). This is the 50th composition in standard order, so a(813) = 50.

Gus Wiseman, Statistics, classes, and transformations of standard compositions.
Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Positions of elements of A233564 are A374767, counted by A374761.
Positions of elements of A272919 are A374759, counted by A374760.
Ranks of rows of A374757 (row-sums A374758).
The weak opposite version is A375123.
The weak version is A375124.
The opposite version is A375125.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1) = A070939(n).
- Parts are listed by A066099.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Run-length transform is A333627.
- Run-compression transform is A373948, sum A373953, excess A373954.
- Ranks of contiguous compositions are A374249, counted by A274174.
- Run-sum transformation is A353847.
Six types of runs:
- Count: A124766, A124765, A124768, A124769, A333381, A124767.
- Leaders: A374629, A374740, A374683, A374757, A374515, A374251.
- Rank: A375123, A375124, A375125, A375126, A375127, A373948.
Cf. A065120, A106356, A188920, A238343, A333213, A373949, A374634, A374685, A374744, A374766, A375128.

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],Greater]],{n,0,100}]

A000120(a(n)) = A124769(n).
A065120(a(n)) = A065120(n).
A070939(a(n)) = A374758(n).

A375127 The anti-run-leader transformation for standard compositions.

Original entry on oeis.org

0, 1, 2, 3, 4, 2, 1, 7, 8, 4, 10, 5, 1, 1, 3, 15, 16, 8, 4, 9, 2, 10, 2, 11, 1, 1, 6, 3, 3, 3, 7, 31, 32, 16, 8, 17, 36, 4, 4, 19, 2, 2, 42, 21, 2, 2, 5, 23, 1, 1, 1, 3, 1, 6, 1, 7, 3, 3, 14, 7, 7, 7, 15, 63, 64, 32, 16, 33, 8, 8, 8, 35, 4, 36, 18, 9, 4, 4, 9

Offset: 0

Gus Wiseman, Aug 02 2024

nonn

The a(n)-th composition in standard order lists the leaders of anti-runs of the n-th composition in standard order.
The leaders of anti-runs in a sequence are obtained by splitting it into maximal consecutive anti-runs (sequences with no adjacent equal terms) and taking the first term of each.
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.
Does this sequence contain all nonnegative integers?

			The 346th composition in standard order is (2,2,1,2,2), with anti-runs ((2),(2,1,2),(2)), with leaders (2,2,2). This is the 42nd composition in standard order, so a(346) = 42.

Gus Wiseman, Statistics, classes, and transformations of standard compositions.
Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Positions of elements of A233564 are A374638, counted by A374518.
Positions of elements of A272919 are A374519, counted by A374517.
Ranks of rows of A374515.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1) = A070939(n).
- Parts are listed by A066099.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Run-length transform is A333627.
- Run-compression transform is A373948, sum A373953, excess A373954.
- Ranks of contiguous compositions are A374249, counted by A274174.
- Run-sum transform is A353847.
Six types of runs:
- Count: A124766, A124765, A124768, A124769, A333381, A124767.
- Leaders: A374629, A374740, A374683, A374757, A374515, A374251.
- Rank: A375123, A375124, A375125, A375126, A375127, A373948.
Cf. A065120, A106356, A188920, A238343, A333213, A373949, A374516, A374521, A374685, A374698, A374701, A374767.

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],UnsameQ]],{n,0,100}]

A000120(a(n)) = A333381(n).
A065120(a(n)) = A065120(n).
A070939(a(n)) = A374516(n).

A375136 Number of maximal strictly increasing runs in the weakly increasing prime factors of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 04 2024

nonn

For n > 1, this is one more than the number of adjacent equal terms in the multiset of prime factors of n.

			The prime factors of 540 are {2,2,3,3,3,5}, with maximal strictly increasing runs ({2},{2,3},{3},{3,5}), so a(540) = 4.

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

For compositions we have A124768, row-lengths of A374683, sum A374684.
For sum of prime indices we have A374706.
Row-lengths of A375128.
A112798 lists prime indices:
- distinct A001221
- length A001222
- leader A055396
- sum A056239
- reverse A296150
Cf. A034296, A046660, A066328, A141199, A141809, A279790, A320324, A333213, A358836, A374700.

Table[Length[Split[Flatten[ConstantArray@@@FactorInteger[n]],Less]],{n,100}]

For n > 1, a(n) = A046660(n) + 1 = A001222(n) - A001221(n) + 1.

A333231 Positions of weak descents in the sequence of differences between primes.

Original entry on oeis.org

2, 4, 6, 9, 11, 12, 15, 16, 18, 19, 21, 24, 25, 27, 30, 32, 34, 36, 37, 39, 40, 42, 44, 46, 47, 48, 51, 53, 54, 55, 56, 58, 59, 62, 63, 66, 68, 72, 73, 74, 77, 80, 82, 84, 87, 88, 91, 92, 94, 97, 99, 101, 102, 103, 106, 107, 108, 110, 111, 112, 114, 115, 118

Offset: 1

Gus Wiseman, Mar 18 2020

Partial sums of A333253.

			The prime gaps split into the following strictly increasing subsequences: (1,2), (2,4), (2,4), (2,4,6), (2,6), (4), (2,4,6), (6), (2,6), (4), (2,6), (4,6,8), (4), (2,4), (2,4,14), ...

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Thomas Bloom, Let d_n = p_(n+1)-p_n. The set of n such that d_(n+1) >= d_n has density 1/2, and similarly for d_(n+1) <= d_n. Furthermore, there are infinitely many n such that d_(n+1) = d_n, Erdős Problems.
Terence Tao, Erdős problem database, see no. 218.

The version for the Kolakoski sequence is A025505.
The version for equal differences is A064113.
The version for strict ascents is A258025.
The version for strict descents is A258026.
The version for distinct differences is A333214.
The version for weak ascents is A333230.
First differences are A333253 (if the first term is 0).
Prime gaps are A001223.
Weakly decreasing runs of compositions in standard order are A124765.
Strictly increasing runs of compositions in standard order are A124768.
Runs of prime gaps with nonzero differences are A333216.
Cf. A000040, A000720, A001221, A036263, A054819, A084758, A114994, A124760, A124761, A124766, A124769, A333212.

Accumulate[Length/@Split[Differences[Array[Prime,100]],#1<#2&]]//Most
- or -
Select[Range[100],Prime[#+1]-Prime[#]>=Prime[#+2]-Prime[#+1]&]

Numbers k such that prime(k+2) - 2*prime(k+1) + prime(k) >= 0.

A374741 Sum of leaders of weakly decreasing runs in the n-th composition in standard order.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Jul 24 2024

nonn

The leaders of weakly decreasing runs in a sequence are obtained by splitting it into maximal weakly decreasing subsequences and taking the first term of each.

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 maximal weakly decreasing subsequences of the 1234567th composition in standard order are ((3,2,1),(2,2,1),(2),(5,1,1,1)), so a(1234567) is 3+2+2+5 = 12.

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

For length instead of sum we have A124765.
Other types of runs are A373953, A374516, A374684, A374758.
The opposite is A374630.
Row-sums of A374740, opposite A374629.
Counting compositions by this statistic gives A374748, opposite A374637.
A373949 counts compositions by run-compressed sum.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1) (or sometimes A070939).
- Parts are listed by A066099.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Number of max runs: A124765, A124766, A124767, A124768, A124769, A333381.
- Ranks of strict compositions are A233564, counted by A032020.
- Constant compositions are ranked by A272919.
- Ranks of anti-run compositions are A333489, counted by A003242.
- Run-length transform is A333627.
- Run-compression transform is A373948.
Cf. A065120, A106356, A188920, A189076, A238343, A333175, A333213, A373949, A373954, A374631, A374633, A374635, A374701, A374742.

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

cp's OEIS Frontend

A333230 Positions of weak ascents in the sequence of differences between primes.

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A374744 Numbers k such that the leaders of weakly decreasing runs in the k-th composition in standard order (A066099) are identical.

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A333215 Lengths of maximal weakly increasing subsequences in the sequence of prime gaps (A001223).

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A375124 Weakly decreasing run-leader transformation for standard compositions.

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A375125 Strictly increasing run-leader transformation for standard compositions.

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A375126 Strictly decreasing run-leader transformation for standard compositions.

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A375127 The anti-run-leader transformation for standard compositions.

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A375136 Number of maximal strictly increasing runs in the weakly increasing prime factors of n.

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