A333216 -id:A333216 - cp's OEIS frontend

A333212 Lengths of maximal weakly decreasing subsequences in the sequence of prime gaps (A001223).

1, 2, 2, 2, 1, 2, 3, 1, 3, 3, 2, 1, 3, 2, 1, 2, 2, 2, 3, 3, 2, 2, 4, 1, 2, 5, 3, 1, 3, 1, 2, 2, 1, 1, 4, 1, 2, 1, 2, 2, 2, 1, 3, 1, 3, 2, 1, 2, 2, 4, 1, 4, 4, 3, 1, 3, 2, 1, 1, 2, 5, 3, 2, 2, 2, 2, 2, 1, 3, 1, 3, 1, 2, 1, 3, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 1, 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 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), ...

First differences of A258025 (with zero prepended).
The version for the Kolakoski sequence is A332273.
The weakly increasing version is A333215.
The unequal version is A333216.
The strictly decreasing version is A333252.
The strictly increasing version is A333253.
The equal version is A333254.
Prime gaps are A001223.
Positions of adjacent equal differences are A064113.
Weakly decreasing runs of compositions in standard order are A124765.
Positions of strict ascents in the sequence of prime gaps are A258025.
Cf. A000040, A000720, A001221, A036263, A054819, A084758, A114994, A124760, A124761, A124768, A333213, A333214.

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

Ones correspond to weak prime quartets A054819, so the sum of terms up to but not including the n-th one is A000720(A054819(n - 1)).

A333253 Lengths of maximal strictly increasing subsequences in the sequence of prime gaps (A001223).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 18 2020

nonn

Prime gaps are differences between adjacent prime numbers.

			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), ...

Wikipedia, Longest increasing subsequence

The weakly decreasing version is A333212.
The weakly increasing version is A333215.
The unequal version is A333216.
First differences of A333231 (if its first term is 0).
The strictly decreasing version is A333252.
The equal version is A333254.
Prime gaps are A001223.
Strictly increasing runs of compositions in standard order are A124768.
Positions of strict ascents in the sequence of prime gaps are A258025.
Cf. A000040, A054819, A064113, A084758, A333214, A333230, A333255.

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

Partial sums are A333231. The partial sum up to but not including the n-th one is A333382(n).

A333252 Lengths of maximal strictly decreasing subsequences in the sequence of prime gaps (A001223).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 18 2020

nonn

Prime gaps are differences between adjacent prime numbers.

			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), (4,2), (4), (14,4), (6,2), (10,2), (6), (6,4), (6), ...

Wikipedia, Longest increasing subsequence

The weakly decreasing version is A333212.
The weakly increasing version is A333215.
The unequal version is A333216.
First differences of A333230 (if the first term is 0).
The strictly increasing version is A333253.
The equal version is A333254.
Prime gaps are A001223.
Strictly decreasing runs of compositions in standard order are A124769.
Positions of strict descents in the sequence of prime gaps are A258026.
Cf. A000040, A064113, A084758, A124764, A124766, A258025, A333213, A333214, A333252, A333256.

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

Partial sums are A333230. The partial sum up to but not including the n-th one is A333381(n - 1).

A333490 First index of unequal prime quartets.

Original entry on oeis.org

7, 8, 10, 11, 13, 17, 18, 19, 20, 22, 23, 24, 28, 30, 31, 32, 34, 40, 42, 44, 47, 49, 50, 51, 52, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 75, 76, 78, 79, 82, 83, 85, 86, 87, 89, 90, 91, 94, 95, 96, 97, 98, 99, 104, 111, 112, 113, 114, 115, 116, 119

Offset: 1

Gus Wiseman, May 15 2020

nonn

Let g(i) = prime(i + 1) - prime(i). These are numbers k such that g(k), g(k + 1), and g(k + 2) are all different.

			The first 10 unequal prime quartets:
  17  19  23  29
  19  23  29  31
  29  31  37  41
  31  37  41  43
  41  43  47  53
  59  61  67  71
  61  67  71  73
  67  71  73  79
  71  73  79  83
  79  83  89  97
For example, 83 is the 23rd prime, and the primes (83,89,97,101) have differences (6,8,4), which are all distinct, so 23 is in the sequence.

Primes are A000040.
Prime gaps are A001223.
Second prime gaps are A036263.
Indices of unequal rows of A066099 are A233564.
Lengths of maximal anti-run subsequences of prime gaps are A333216.
Lengths of maximal runs of prime gaps are A333254.
Maximal anti-runs in standard compositions are counted by A333381.
Indices of anti-run rows of A066099 are A333489.
Strictly decreasing prime quartets are A054804.
Strictly increasing prime quartets are A054819.
Equal prime quartets are A090832.
Weakly increasing prime quartets are A333383.
Weakly decreasing prime quartets are A333488.
Unequal prime quartets are A333490 (this sequence).
Partially unequal prime quartets are A333491.
Positions of adjacent equal prime gaps are A064113.
Positions of strict ascents in prime gaps are A258025.
Positions of strict descents in prime gaps are A258026.
Positions of adjacent unequal prime gaps are A333214.
Positions of weak ascents in prime gaps are A333230.
Positions of weak descents in prime gaps are A333231.
Cf. A006560, A031217, A054800, A059044, A084758, A089180, A124767, A333215.

ReplaceList[Array[Prime,100],{_,x_,y_,z_,t_,_}/;y-x!=z-y!=t-z:>PrimePi[x]]

A333491 First index of partially unequal prime quartets.

Original entry on oeis.org

3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 37, 40, 41, 42, 43, 44, 47, 48, 49, 50, 51, 52, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 74, 75, 76, 77, 78, 79, 80, 81, 82

Offset: 1

Gus Wiseman, May 15 2020

nonn

Let g(i) = prime(i + 1) - prime(i). These are numbers k such that g(k) != g(k + 1) != g(k + 2), but we may have g(k) = g(k + 2).

			The first 10 partially unequal prime quartets:
   5  7 11 13
   7 11 13 17
  11 13 17 19
  13 17 19 23
  17 19 23 29
  19 23 29 31
  23 29 31 37
  29 31 37 41
  31 37 41 43
  37 41 43 47

Primes are A000040.
Prime gaps are A001223.
Second prime gaps are A036263.
Indices of unequal rows of A066099 are A233564.
Lengths of maximal anti-runs of prime gaps are A333216.
Lengths of maximal runs of prime gaps are A333254.
Maximal anti-runs in standard compositions are counted by A333381.
Indices of anti-run rows of A066099 are A333489.
Strictly decreasing prime quartets are A054804.
Strictly increasing prime quartets are A054819.
Equal prime quartets are A090832.
Weakly increasing prime quartets are A333383.
Weakly decreasing prime quartets are A333488.
Unequal prime quartets are A333490.
Partially unequal prime quartets are A333491 (this sequence).
Positions of adjacent equal prime gaps are A064113.
Positions of strict ascents in prime gaps are A258025.
Positions of strict descents in prime gaps are A258026.
Positions of adjacent unequal prime gaps are A333214.
Positions of weak ascents in prime gaps are A333230.
Positions of weak descents in prime gaps are A333231.
Cf. A006560, A031217, A054800, A059044, A084758, A089180, A124767, A333215.

ReplaceList[Array[Prime,100],{_,x_,y_,z_,t_,_}/;y-x!=z-y&&z-y!=t-z:>PrimePi[x]]
PrimePi[#]&/@(Select[Partition[Prime[Range[90]],4,1],#[[2]]-#[[1]]!=#[[3]]-#[[2]]&&#[[3]]-#[[2]]!=#[[4]]-#[[3]]&][[;;,1]]) (* Harvey P. Dale, Aug 05 2025 *)

A337565 Irregular triangle read by rows where row k is the sequence of maximal anti-run lengths in the k-th composition in standard order.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Sep 07 2020

An anti-run is a sequence with no adjacent equal parts.

A composition of n is a finite sequence of positive integers summing to n. 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 first column below lists various selected n; the second column gives the corresponding composition; the third column gives the corresponding row of the triangle, i.e., the anti-run lengths.
    1:           (1) -> (1)
    3:         (1,1) -> (1,1)
    5:         (2,1) -> (2)
    7:       (1,1,1) -> (1,1,1)
   11:       (2,1,1) -> (2,1)
   13:       (1,2,1) -> (3)
   14:       (1,1,2) -> (1,2)
   15:     (1,1,1,1) -> (1,1,1,1)
   23:     (2,1,1,1) -> (2,1,1)
   27:     (1,2,1,1) -> (3,1)
   29:     (1,1,2,1) -> (1,3)
   30:     (1,1,1,2) -> (1,1,2)
   31:   (1,1,1,1,1) -> (1,1,1,1,1)
   43:     (2,2,1,1) -> (1,2,1)
   45:     (2,1,2,1) -> (4)
   46:     (2,1,1,2) -> (2,2)
   47:   (2,1,1,1,1) -> (2,1,1,1)
   55:   (1,2,1,1,1) -> (3,1,1)
   59:   (1,1,2,1,1) -> (1,3,1)
   61:   (1,1,1,2,1) -> (1,1,3)
   62:   (1,1,1,1,2) -> (1,1,1,2)
   63: (1,1,1,1,1,1) -> (1,1,1,1,1,1)
For example, the 119th composition is (1,1,2,1,1,1), with maximal anti-runs ((1),(1,2,1),(1),(1)), so row 119 is (1,3,1,1).

A000120 gives row sums.
A333381 gives row lengths.
A333769 is the version for runs.
A003242 counts anti-run compositions.
A011782 counts compositions.
A106351 counts anti-run compositions by length.
A329744 is a triangle counting compositions by runs-resistance.
A333755 counts compositions by number of runs.
All of the following pertain to compositions in standard order (A066099):
- Sum is A070939.
- Adjacent equal pairs are counted by A124762.
- Runs are counted by A124767.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Patterns are A333217.
- Heinz number is A333219.
- Anti-runs are counted by A333381.
- Anti-run compositions are A333489.
- Runs-resistance is A333628.
- Combinatory separations are A334030.
Cf. A106356, A113835, A114994, A124767, A181819, A228351, A238279, A318928, A333216, A333627, A333630.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length/@Split[stc[n],UnsameQ],{n,0,50}]

A337504 Number of compositions of 2*n with n maximal anti-runs.

Original entry on oeis.org

1, 1, 3, 8, 13, 33, 112, 286, 769, 2288, 6695, 18745, 54654, 160888, 467402, 1362378, 4016517, 11807966, 34708018, 102451390, 302870005, 895207191, 2650590597, 7859253320, 23316653154, 69231883374, 205773157904, 612021943421, 1821435719846, 5424528040529, 16165017705176

Offset: 0

Gus Wiseman, Sep 04 2020

nonn

An anti-run is a sequence with no adjacent equal parts.

			The a(0) = 1 through a(4) = 13 compositions:
  ()  (2)  (2,2)    (2,2,2)      (2,2,2,2)
           (1,1,2)  (1,1,1,3)    (1,1,1,1,4)
           (2,1,1)  (1,1,2,2)    (1,1,2,2,2)
                    (2,2,1,1)    (2,2,2,1,1)
                    (3,1,1,1)    (4,1,1,1,1)
                    (1,1,1,2,1)  (1,1,1,1,3,1)
                    (1,1,2,1,1)  (1,1,1,2,2,1)
                    (1,2,1,1,1)  (1,1,1,3,1,1)
                                 (1,1,2,2,1,1)
                                 (1,1,3,1,1,1)
                                 (1,2,2,1,1,1)
                                 (1,3,1,1,1,1)
                                 (2,1,1,1,1,2)

Andrew Howroyd, Table of n, a(n) for n = 0..200

A106356 has this as main diagonal n = 2*k.
A336108 is the version for runs.
A337505 is the version for patterns.
A337564 is the version for runs in patterns.
A003242 counts anti-run compositions.
A011782 counts compositions.
A124767 counts runs in standard compositions.
A238343 counts compositions by descents.
A333213 counts compositions by weak ascents.
A333381 counts anti-runs in standard compositions.
A333382 counts adjacent unequal pairs in standard compositions.
A333489 ranks anti-runs.
A333755 counts compositions by number of runs.
A333769 gives run-lengths in standard compositions.
A337565 gives anti-run lengths in standard compositions.
Cf. A106351, A124762, A233564, A235998, A238130, A238279, A333214, A333216.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[2*n],Length[Split[#,UnsameQ]]==n&]],{n,0,10}]

a(n)={polcoef(polcoef(1 - y + y*(y-1)/(y - 1 - sum(d=1, 2*n, (y-1)^d*x^d/(1 - x^d) + O(x^(2*n+1)))), 2*n, x), n, y)} \\ Andrew Howroyd, Feb 02 2021

a(n) = [x^(2*n)*y^n] 1 - y + y*(y-1)/(y - 1 - Sum_{d>=1} (y-1)^d*x^d/(1 - x^d)). - Andrew Howroyd, Feb 02 2021

Terms a(11) and beyond from Andrew Howroyd, Feb 02 2021

cp's OEIS Frontend

A333212 Lengths of maximal weakly decreasing subsequences in the sequence of prime gaps (A001223).

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A333253 Lengths of maximal strictly increasing subsequences in the sequence of prime gaps (A001223).

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A333252 Lengths of maximal strictly decreasing subsequences in the sequence of prime gaps (A001223).

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A333490 First index of unequal prime quartets.

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A333491 First index of partially unequal prime quartets.

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A337565 Irregular triangle read by rows where row k is the sequence of maximal anti-run lengths in the k-th composition in standard order.

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A337504 Number of compositions of 2*n with n maximal anti-runs.

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