A333218 -id:A333218 - cp's OEIS frontend

A335483 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (3,1,2).

38, 70, 77, 78, 102, 134, 140, 141, 142, 150, 154, 155, 157, 158, 166, 198, 205, 206, 230, 262, 268, 269, 270, 276, 278, 281, 282, 283, 284, 285, 286, 294, 301, 302, 306, 308, 309, 310, 311, 314, 315, 317, 318, 326, 333, 334, 358, 390, 396, 397, 398, 406, 410

Offset: 1

Gus Wiseman, Jun 18 2020

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 a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

			The sequence of terms together with the corresponding compositions begins:
   38: (3,1,2)
   70: (4,1,2)
   77: (3,1,2,1)
   78: (3,1,1,2)
  102: (1,3,1,2)
  134: (5,1,2)
  140: (4,1,3)
  141: (4,1,2,1)
  142: (4,1,1,2)
  150: (3,2,1,2)
  154: (3,1,2,2)
  155: (3,1,2,1,1)
  157: (3,1,1,2,1)
  158: (3,1,1,1,2)
  166: (2,3,1,2)

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.
Gus Wiseman, Statistics, classes, and transformations of standard compositions

The version counting permutations is A056986.
Patterns matching this pattern are counted by A335515 (by length).
Permutations of prime indices matching this pattern are counted by A335520.
These compositions are counted by A335514 (by sum).
Constant patterns are counted by A000005 and ranked by A272919.
Permutations are counted by A000142 and ranked by A333218.
Patterns are counted by A000670 and ranked by A333217.
Non-unimodal compositions are counted by A115981 and ranked by A335373.
Permutations matching (1,3,2,4) are counted by A158009.
Combinatory separations are counted by A269134.
Patterns matched by standard compositions are counted by A335454.
Minimal patterns avoided by a standard composition are counted by A335465.
Other permutations:
- A335479 (1,2,3)
- A335480 (1,3,2)
- A335481 (2,1,3)
- A335482 (2,3,1)
- A335483 (3,1,2)
- A335484 (3,2,1)
Cf. A034691, A056986, A108917, A114994, A158005, A238279, A333224, A333257, A334968, A335456, A335458.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
Select[Range[0,100],MatchQ[stc[#],{_,x_,_,y_,_,z_,_}/;y

A335486 Numbers k such that the k-th composition in standard order (A066099) is not weakly increasing.

Original entry on oeis.org

5, 9, 11, 13, 17, 18, 19, 21, 22, 23, 25, 27, 29, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 59, 61, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 97, 98, 99

Offset: 1

Gus Wiseman, Jun 18 2020

nonn

Also compositions matching the pattern (2,1).

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 sequence of terms together with the corresponding compositions begins:
   5: (2,1)
   9: (3,1)
  11: (2,1,1)
  13: (1,2,1)
  17: (4,1)
  18: (3,2)
  19: (3,1,1)
  21: (2,2,1)
  22: (2,1,2)
  23: (2,1,1,1)
  25: (1,3,1)
  27: (1,2,1,1)
  29: (1,1,2,1)
  33: (5,1)
  34: (4,2)
  35: (4,1,1)

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.
Gus Wiseman, Statistics, classes, and transformations of standard compositions

The complement A225620 is the avoiding version.
The (1,2)-matching version is A335485.
Patterns matching this pattern are counted by A002051 (by length).
Permutations of prime indices matching this pattern are counted by A008480(n) - 1.
These compositions are counted by A056823 (by sum).
Constant patterns are counted by A000005 and ranked by A272919.
Permutations are counted by A000142 and ranked by A333218.
Patterns are counted by A000670 and ranked by A333217.
Non-unimodal compositions are counted by A115981 and ranked by A335373.
Combinatory separations are counted by A269134.
Patterns matched by standard compositions are counted by A335454.
Minimal patterns avoided by a standard composition are counted by A335465.
Cf. A034691, A056986, A108917, A114994, A238279, A334968, A335456, A335458.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
Select[Range[0,100],MatchQ[stc[#],{_,x_,_,y_,_}/;x>y]&]

A333380 Numbers k such that the k-th composition in standard order is weakly decreasing and covers an initial interval of positive integers.

Original entry on oeis.org

0, 1, 3, 5, 7, 11, 15, 21, 23, 31, 37, 43, 47, 63, 75, 85, 87, 95, 127, 149, 151, 171, 175, 191, 255, 293, 299, 303, 341, 343, 351, 383, 511, 549, 587, 597, 599, 607, 683, 687, 703, 767, 1023, 1099, 1173, 1175, 1195, 1199, 1215, 1365, 1367, 1375, 1407, 1535

Offset: 1

Gus Wiseman, Mar 21 2020

nonn

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of 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.

			The sequence of terms together with the corresponding compositions begins:
    0: ()               127: (1,1,1,1,1,1,1)
    1: (1)              149: (3,2,2,1)
    3: (1,1)            151: (3,2,1,1,1)
    5: (2,1)            171: (2,2,2,1,1)
    7: (1,1,1)          175: (2,2,1,1,1,1)
   11: (2,1,1)          191: (2,1,1,1,1,1,1)
   15: (1,1,1,1)        255: (1,1,1,1,1,1,1,1)
   21: (2,2,1)          293: (3,3,2,1)
   23: (2,1,1,1)        299: (3,2,2,1,1)
   31: (1,1,1,1,1)      303: (3,2,1,1,1,1)
   37: (3,2,1)          341: (2,2,2,2,1)
   43: (2,2,1,1)        343: (2,2,2,1,1,1)
   47: (2,1,1,1,1)      351: (2,2,1,1,1,1,1)
   63: (1,1,1,1,1,1)    383: (2,1,1,1,1,1,1,1)
   75: (3,2,1,1)        511: (1,1,1,1,1,1,1,1,1)
   85: (2,2,2,1)        549: (4,3,2,1)
   87: (2,2,1,1,1)      587: (3,3,2,1,1)
   95: (2,1,1,1,1,1)    597: (3,2,2,2,1)

Sequences covering an initial interval are counted by A000670.
Compositions in standard order are A066099.
Weakly decreasing runs are counted by A124765.
Removing the covering condition gives A114994.
Removing the ordering condition gives A333217.
The strictly decreasing case is A246534.
The unequal version is A333218.
The weakly increasing version is A333379.
Cf. A000120, A000225, A029931, A048793, A070939, A164894, A225620, A228351, A233564, A272919, A333219.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,1000],normQ[stc[#]]&&GreaterEqual@@stc[#]&]

Intersection of A333217 and A114994.

A335468 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (2,1,2).

Original entry on oeis.org

22, 45, 46, 54, 76, 86, 90, 91, 93, 94, 109, 110, 118, 148, 150, 153, 156, 166, 173, 174, 178, 180, 181, 182, 183, 186, 187, 189, 190, 204, 214, 218, 219, 221, 222, 237, 238, 246, 278, 280, 297, 300, 301, 302, 306, 307, 308, 310, 313, 316, 326, 332, 333, 334

Offset: 1

Gus Wiseman, Jun 16 2020

nonn

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.

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

			The sequence together with the corresponding compositions begins:
   22: (2,1,2)
   45: (2,1,2,1)
   46: (2,1,1,2)
   54: (1,2,1,2)
   76: (3,1,3)
   86: (2,2,1,2)
   90: (2,1,2,2)
   91: (2,1,2,1,1)
   93: (2,1,1,2,1)
   94: (2,1,1,1,2)
  109: (1,2,1,2,1)
  110: (1,2,1,1,2)
  118: (1,1,2,1,2)
  148: (3,2,3)
  150: (3,2,1,2)

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.
Gus Wiseman, Statistics, classes, and transformations of standard compositions

The complement A335469 is the avoiding version.
The (1,2,1)-matching version is A335466.
These compositions are counted by A335472.
Constant patterns are counted by A000005 and ranked by A272919.
Permutations are counted by A000142 and ranked by A333218.
Patterns are counted by A000670 and ranked by A333217.
Non-unimodal compositions are counted by A115981 and ranked by A335373.
Combinatory separations are counted by A269134 and ranked by A334030.
Patterns matched by standard compositions are counted by A335454.
Minimal patterns avoided by a standard composition are counted by A335465.
Cf. A034691, A056986, A108917, A114994, A238279, A333224, A333257, A335453, A335456, A335458, A335509.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
Select[Range[0,100],MatchQ[stc[#],{_,x_,_,y_,_,x_,_}/;x>y]&];

A335481 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (2,1,3).

Original entry on oeis.org

44, 88, 89, 92, 108, 152, 172, 176, 177, 178, 179, 180, 184, 185, 188, 216, 217, 220, 236, 296, 300, 304, 305, 312, 332, 344, 345, 348, 352, 353, 354, 355, 356, 357, 358, 359, 360, 361, 364, 368, 369, 370, 371, 372, 376, 377, 380, 408, 428, 432, 433, 434, 435

Offset: 1

Gus Wiseman, Jun 18 2020

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 a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

			The sequence of terms together with the corresponding compositions begins:
   44: (2,1,3)
   88: (2,1,4)
   89: (2,1,3,1)
   92: (2,1,1,3)
  108: (1,2,1,3)
  152: (3,1,4)
  172: (2,2,1,3)
  176: (2,1,5)
  177: (2,1,4,1)
  178: (2,1,3,2)
  179: (2,1,3,1,1)
  180: (2,1,2,3)
  184: (2,1,1,4)
  185: (2,1,1,3,1)
  188: (2,1,1,1,3)

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.
Gus Wiseman, Statistics, classes, and transformations of standard compositions

The version counting permutations is A056986.
Patterns matching this pattern are counted by A335515 (by length).
Permutations of prime indices matching this pattern are counted by A335520.
These compositions are counted by A335514 (by sum).
Constant patterns are counted by A000005 and ranked by A272919.
Permutations are counted by A000142 and ranked by A333218.
Patterns are counted by A000670 and ranked by A333217.
Non-unimodal compositions are counted by A115981 and ranked by A335373.
Permutations matching (1,3,2,4) are counted by A158009.
Combinatory separations are counted by A269134.
Patterns matched by standard compositions are counted by A335454.
Minimal patterns avoided by a standard composition are counted by A335465.
Other permutations:
- A335479 (1,2,3)
- A335480 (1,3,2)
- A335481 (2,1,3)
- A335482 (2,3,1)
- A335483 (3,1,2)
- A335484 (3,2,1)
Cf. A034691, A056986, A108917, A114994, A158005, A238279, A333224, A333257, A334968, A335456, A335458.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
Select[Range[0,100],MatchQ[stc[#],{_,x_,_,y_,_,z_,_}/;y

A335484 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (3,2,1).

Original entry on oeis.org

37, 69, 75, 77, 101, 133, 137, 139, 141, 149, 150, 151, 155, 157, 165, 197, 203, 205, 229, 261, 265, 267, 269, 274, 275, 277, 278, 279, 281, 283, 285, 293, 297, 299, 300, 301, 302, 303, 309, 310, 311, 315, 317, 325, 331, 333, 357, 389, 393, 395, 397, 405, 406

Offset: 1

Gus Wiseman, Jun 18 2020

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 a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

			The sequence of terms together with the corresponding compositions begins:
   37: (3,2,1)
   69: (4,2,1)
   75: (3,2,1,1)
   77: (3,1,2,1)
  101: (1,3,2,1)
  133: (5,2,1)
  137: (4,3,1)
  139: (4,2,1,1)
  141: (4,1,2,1)
  149: (3,2,2,1)
  150: (3,2,1,2)
  151: (3,2,1,1,1)
  155: (3,1,2,1,1)
  157: (3,1,1,2,1)
  165: (2,3,2,1)

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.
Gus Wiseman, Statistics, classes, and transformations of standard compositions

The version counting permutations is A056986.
Patterns matching this pattern are counted by A335515 (by length).
Permutations of prime indices matching this pattern are counted by A335520.
These compositions are counted by A335514 (by sum).
Constant patterns are counted by A000005 and ranked by A272919.
Permutations are counted by A000142 and ranked by A333218.
Patterns are counted by A000670 and ranked by A333217.
Non-unimodal compositions are counted by A115981 and ranked by A335373.
Permutations matching (1,3,2,4) are counted by A158009.
Combinatory separations are counted by A269134.
Patterns matched by standard compositions are counted by A335454.
Minimal patterns avoided by a standard composition are counted by A335465.
Other permutations:
- A335479 (1,2,3)
- A335480 (1,3,2)
- A335481 (2,1,3)
- A335482 (2,3,1)
- A335483 (3,1,2)
- A335484 (3,2,1)
Cf. A034691, A056986, A108917, A114994, A158005, A238279, A333224, A333257, A334968, A335456, A335458.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
Select[Range[0,100],MatchQ[stc[#],{_,x_,_,y_,_,z_,_}/;z

A335524 Numbers k such that the k-th composition in standard order (A066099) avoids the pattern (2,2,1).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 46, 47, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71

Offset: 1

Gus Wiseman, Jun 18 2020

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 a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.
Gus Wiseman, Statistics, classes, and transformations of standard compositions

Patterns avoiding this pattern are counted by A001710 (by length).
Permutations of prime indices avoiding this pattern are counted by A335450.
These compositions are counted by A335473 (by sum).
The complement A335477 is the matching version.
The (1,2,2)-avoiding version is A335525.
Constant patterns are counted by A000005 and ranked by A272919.
Permutations are counted by A000142 and ranked by A333218.
Patterns are counted by A000670 and ranked by A333217.
Non-unimodal compositions are counted by A115981 and ranked by A335373.
Combinatory separations are counted by A269134.
Patterns matched by standard compositions are counted by A335454.
Minimal patterns avoided by a standard composition are counted by A335465.
Cf. A034691, A056986, A108917, A114994, A238279, A333224, A333257, A334968, A335446, A335456, A335458.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
Select[Range[0,100],!MatchQ[stc[#],{_,x_,_,x_,_,y_,_}/;x>y]&]

A335525 Numbers k such that the k-th composition in standard order (A066099) avoids the pattern (1,2,2).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71

Offset: 1

Gus Wiseman, Jun 18 2020

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 a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.
Gus Wiseman, Statistics, classes, and transformations of standard compositions

Patterns avoiding this pattern are counted by A001710 (by length).
Permutations of prime indices avoiding this pattern are counted by A335450.
These compositions are counted by A335473 (by sum).
The complement A335475 is the matching version.
The (2,2,1)-avoiding version is A335524.
Constant patterns are counted by A000005 and ranked by A272919.
Permutations are counted by A000142 and ranked by A333218.
Patterns are counted by A000670 and ranked by A333217.
Non-unimodal compositions are counted by A115981 and ranked by A335373.
Combinatory separations are counted by A269134.
Patterns matched by standard compositions are counted by A335454.
Minimal patterns avoided by a standard composition are counted by A335465.
Cf. A034691, A056986, A108917, A114994, A238279, A333224, A333257, A334968, A335446, A335456, A335458.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
Select[Range[0,100],!MatchQ[stc[#],{_,x_,_,y_,_,y_,_}/;x

A334032 The a(n)-th composition in standard order (graded reverse-lexicographic) is the unsorted prime signature of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 5, 1, 3, 3, 8, 1, 6, 1, 5, 3, 3, 1, 9, 2, 3, 4, 5, 1, 7, 1, 16, 3, 3, 3, 10, 1, 3, 3, 9, 1, 7, 1, 5, 5, 3, 1, 17, 2, 6, 3, 5, 1, 12, 3, 9, 3, 3, 1, 11, 1, 3, 5, 32, 3, 7, 1, 5, 3, 7, 1, 18, 1, 3, 6, 5, 3, 7, 1, 17, 8, 3, 1, 11

Offset: 1

Gus Wiseman, Apr 17 2020

nonn

Unsorted prime signature (A124010) is the sequence of exponents in a number's prime factorization.

The k-th composition in standard order (row k of 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 unsorted prime signature of 12345678 is (1,2,1,1), which is the 27th composition in standard order, so a(12345678) = 27.

Positions of first appearances are A057335 (a partial inverse).
Least number with same prime signature is A071364.
Unsorted prime signature is A124010.
Least number with reversed prime signature is A331580.
Minimal numbers with standard reversed prime signatures are A334031.
The reversed version is A334033.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Sum is A070939.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Aperiodic compositions are A328594.
- Normal compositions are A333217.
- Permutations are A333218.
- Heinz number is A333219.
Cf. A029931, A048793, A052409, A055932, A056239, A112798, A124767, A228351, A233249, A329139, A333220.

stcinv[q_]:=Total[2^Accumulate[Reverse[q]]]/2;
Table[stcinv[Last/@If[n==1,{},FactorInteger[n]]],{n,100}]

a(A057335(n)) = n.
A057335(a(n)) = A071364(n).
a(A334031(n))= A059893(n).
A334031(a(n)) = A331580(n).

A335488 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (1,1).

Original entry on oeis.org

3, 7, 10, 11, 13, 14, 15, 19, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 35, 36, 39, 42, 43, 45, 46, 47, 49, 51, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 67, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 97, 99, 100, 101

Offset: 1

Gus Wiseman, Jun 18 2020

nonn

These are compositions with some part appearing more than once, or non-strict compositions.
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.
We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

			The sequence of terms together with the corresponding compositions begins:
   3: (1,1)
   7: (1,1,1)
  10: (2,2)
  11: (2,1,1)
  13: (1,2,1)
  14: (1,1,2)
  15: (1,1,1,1)
  19: (3,1,1)
  21: (2,2,1)
  22: (2,1,2)
  23: (2,1,1,1)
  25: (1,3,1)
  26: (1,2,2)
  27: (1,2,1,1)
  28: (1,1,3)

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.
Gus Wiseman, Statistics, classes, and transformations of standard compositions

The complement A233564 is the avoiding version.
Patterns matching this pattern are counted by A019472 (by length).
Permutations of prime indices matching this pattern are counted by A335487.
These compositions are counted by A261982 (by sum).
Constant patterns are counted by A000005 and ranked by A272919.
Permutations are counted by A000142 and ranked by A333218.
Patterns are counted by A000670 and ranked by A333217.
Non-unimodal compositions are counted by A115981 and ranked by A335373.
Combinatory separations are counted by A269134.
Patterns matched by standard compositions are counted by A335454.
Minimal patterns avoided by a standard composition are counted by A335465.
The (1,1,1)-matching case is A335512.
Cf. A034691, A056986, A108917, A114994, A238279, A333224, A333257, A334968, A335456, A335458.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
Select[Range[0,100],MatchQ[stc[#],{_,x_,_,x_,_}]&]

cp's OEIS Frontend

A335483 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (3,1,2).

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A335486 Numbers k such that the k-th composition in standard order (A066099) is not weakly increasing.

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A333380 Numbers k such that the k-th composition in standard order is weakly decreasing and covers an initial interval of positive integers.

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A335468 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (2,1,2).

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A335481 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (2,1,3).

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A335484 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (3,2,1).

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A335524 Numbers k such that the k-th composition in standard order (A066099) avoids the pattern (2,2,1).

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A335525 Numbers k such that the k-th composition in standard order (A066099) avoids the pattern (1,2,2).

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A334032 The a(n)-th composition in standard order (graded reverse-lexicographic) is the unsorted prime signature of n.