cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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

Views

Author

Gus Wiseman, May 15 2020

Keywords

Comments

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.

Examples

			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.

Crossrefs

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.

Programs

  • Mathematica
    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

Views

Author

Gus Wiseman, May 15 2020

Keywords

Comments

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).

Examples

			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

Crossrefs

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.

Programs

  • Mathematica
    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 *)

A333488 First index of weakly decreasing prime quartets.

Original entry on oeis.org

11, 15, 18, 24, 36, 39, 46, 47, 53, 54, 55, 58, 62, 72, 73, 87, 91, 101, 102, 106, 107, 110, 111, 114, 118, 127, 128, 129, 132, 146, 150, 157, 180, 186, 193, 199, 210, 217, 223, 228, 232, 239, 242, 259, 260, 263, 269, 270, 271, 274, 275, 282, 283, 284, 290
Offset: 1

Views

Author

Gus Wiseman, May 15 2020

Keywords

Comments

Let g(i) = prime(i + 1) - prime(i). These are numbers k such that g(k) >= g(k + 1) >= g(k + 2).

Examples

			The first 10 weakly decreasing prime quartets:
   31  37  41  43
   47  53  59  61
   61  67  71  73
   89  97 101 103
  151 157 163 167
  167 173 179 181
  199 211 223 227
  211 223 227 229
  241 251 257 263
  251 257 263 269
For example, 241 is the 53rd prime, and the primes (241,251,257,263) have differences (10,6,6), which are weakly decreasing, so 53 is in the sequence.

Crossrefs

Prime gaps are A001223.
Second prime gaps are A036263.
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 (this sequence).
Unequal prime quartets are A333490.
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.
Indices of weakly decreasing rows of A066099 are A114994.
Lengths of maximal weakly decreasing subsequences of prime gaps: A333212.
Lengths of maximal strictly increasing subsequences of prime gaps: A333253.
Cf. A000040, A006560, A031217, A054800, A059044, A084758, A089180.

Programs

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

A335277 First index of strictly increasing prime quartets.

Original entry on oeis.org

7, 13, 22, 28, 49, 60, 64, 69, 70, 75, 78, 85, 89, 95, 104, 116, 122, 123, 144, 148, 152, 155, 173, 178, 182, 195, 201, 206, 212, 215, 219, 225, 226, 230, 236, 237, 244, 253, 256, 257, 265, 288, 302, 307, 315, 325, 328, 329, 332, 333, 336, 348, 355, 361, 373
Offset: 1

Views

Author

Gus Wiseman, May 30 2020

Keywords

Comments

Let g(i) = prime(i + 1) - prime(i). These are numbers k such that g(k) < g(k + 1) < g(k + 2).

Examples

			The first 10 strictly increasing prime quartets:
   17  19  23  29
   41  43  47  53
   79  83  89  97
  107 109 113 127
  227 229 233 239
  281 283 293 307
  311 313 317 331
  347 349 353 359
  349 353 359 367
  379 383 389 397
For example, 107 is the 28th prime, and the primes (107,109,113,127) have differences (2,4,14), which are strictly increasing, so 28 is in the sequence.

Crossrefs

Prime gaps are A001223.
Second prime gaps are A036263.
Strictly decreasing prime quartets are A335278.
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.
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.
Lengths of maximal weakly decreasing sequences of prime gaps are A333212.
Lengths of maximal strictly increasing sequences of prime gaps are A333253.
Cf. A000040, A006560, A031217, A054800, A054819, A059044, A084758, A089180.

Programs

  • Mathematica
    ReplaceList[Array[Prime,100],{_,x_,y_,z_,t_,_}/;y-xPrimePi[x]]

Formula

prime(a(n)) = A054819(n).

A335278 First index of strictly decreasing prime quartets.

Original entry on oeis.org

11, 18, 24, 47, 58, 62, 87, 91, 111, 114, 127, 132, 146, 150, 157, 180, 210, 223, 228, 232, 242, 259, 260, 263, 269, 274, 275, 282, 283, 284, 299, 300, 309, 321, 344, 350, 351, 363, 364, 367, 368, 369, 375, 378, 382, 388, 393, 399, 406, 409, 413, 431, 442, 446
Offset: 1

Views

Author

Gus Wiseman, May 30 2020

Keywords

Comments

Let g(i) = prime(i + 1) - prime(i). These are numbers k such that g(k) > g(k + 1) > g(k + 2).

Examples

			The first 10 strictly decreasing prime quartets:
   31  37  41  43
   61  67  71  73
   89  97 101 103
  211 223 227 229
  271 277 281 283
  293 307 311 313
  449 457 461 463
  467 479 487 491
  607 613 617 619
  619 631 641 643
For example, 211 is the 47th prime, and the primes (211,223,227,229) have differences (12,4,2), which are strictly decreasing, so 47 is in the sequence.

Crossrefs

Prime gaps are A001223.
Second prime gaps are A036263.
Strictly increasing prime quartets are A335277.
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.
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.
Indices of strictly decreasing rows of A066099 are A333256.
Lengths of maximal weakly increasing sequences of prime gaps are A333215.
Lengths of maximal strictly decreasing sequences of prime gaps are A333252.
Cf. A000040, A006560, A031217, A054800, A054804, A059044, A084758, A089180, A333253.

Programs

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

Formula

prime(a(n)) = A054804(n).
Previous Showing 11-15 of 15 results.

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