A023192 -id:A023192 - cp's OEIS frontend

A023189 Conjecturally, number of infinitely recurring prime patterns of width 2n-1.

1, 1, 1, 3, 4, 4, 14, 13, 16, 48, 55, 50, 173, 148, 147, 665, 580, 559, 1920, 1447, 1975, 6240, 4228, 5689, 15764, 17562, 14332, 46207, 39071, 35317, 172311, 134752, 110758, 381384, 299971, 479935, 1154568, 733900, 1027967, 2581763, 2636545, 2333308, 8369027, 5516720, 6043194

Offset: 1

David W. Wilson

Of the patterns counted by A023192, the number of those that start and end with a prime. - Sean A. Irvine, May 27 2019

			From _Jon E. Schoenfield_, May 17 2024: (Start)
The table below lists every (conjecturally) infinitely recurring prime pattern of width 2n-1 for n = 1..7. Each p represents a prime; each c represents a composite.
.
  n  2n-1  a(n)  prime patterns
  -  ----  ----  --------------------------------------------------
  1     1     1  p
  2     3     1  pcp
  3     5     1  pcccp
  4     7     3  pcccccp, pcpcccp, pcccpcp
  5     9     4  pcccccccp, pcpcccccp, pcccccpcp, pcpcccpcp
  6    11     4  pcccccccccp, pcccpcccccp, pcccccpcccp, pcccpcpcccp
  7    13    14  pcccccccccccp, pcpcccccccccp, pcccpcccccccp,
                 pcccccpcccccp, pcccccccpcccp, pcccccccccpcp,
                 pcpcccpcccccp, pcpcccccpcccp, pcccpcpcccccp,
                 pcccpcccccpcp, pcccccpcpcccp, pcccccpcccpcp,
                 pcpcccpcpcccp, pcccpcpcccpcp
(End)

Pontus von Brömssen, Table of n, a(n) for n = 1..60

Cf. A023190, A023191, A023192.

Name edited by Jon E. Schoenfield, May 17 2024

a(43)-a(45) from Pontus von Brömssen, Aug 25 2025

A023190 Conjecturally, maximum number of primes in an infinitely-recurring prime pattern of width 2*n-1.

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 5, 5, 6, 6, 7, 6, 7, 8, 8, 9, 10, 10, 11, 10, 11, 12, 12, 12, 13, 14, 13, 14, 15, 15, 16, 16, 16, 17, 17, 18, 18, 18, 19, 19, 20, 20, 21, 21, 21, 22, 22, 23, 23, 23, 24, 24, 24, 24, 24, 25, 25, 26, 26, 26, 27, 27, 27, 28, 28, 29, 29, 29, 30, 30

Offset: 1

David W. Wilson

nonn

Of all the patterns in A023192 (i.e. infinitely-recurring prime patterns) for length 2*n-1, consider those starting and ending with "p". This sequence gives the maximal count of "p"'s in any of those patterns. The companion sequence A023191, gives the number of patterns achieving that maximum. - Sean A. Irvine, May 27 2019

			a(3) concerns patterns of length 5. Of the 10 potential patterns (ccccc, ccccp, cccpc, ccpcc, cpccc, pcccc, ccpcp, cpcpc, pcpcc, pcccp), only pcccp starts and ends with a "p", and it contains 2 "p"'s, so a(3) = 2, and A023191(3) = 1. - _Sean A. Irvine_, May 27 2019

Martin Raab, Table of n, a(n) for n = 1..1166
Thomas J Engelsma, Permissible Patterns

Cf. A023191, A023192.

More terms from Thomas J Engelsma web page added by Martin Raab, Oct 31 2021

A280417 Number of distinct length-n blocks (a.k.a. subword complexity) of the characteristic sequence of the prime numbers A010051.

Original entry on oeis.org

1, 2, 4, 7, 9, 13, 16, 22, 28, 38, 48, 62, 76, 104, 132, 174, 216, 273, 330, 435, 540, 700, 860

Offset: 0

Jeffrey Shallit, Jan 02 2017

Unlike A023192, this sequence also counts blocks that occur finitely often in A010051. And unlike A023192, the correctness of the numbers provided here depend on no conjecture.

			For n = 4, the 9 blocks (in the order they occur in A010051) are 0110,1101,1010,0101,0100,1000,0001,0010,0000.

Cf. A010051, A023192. A280418 gives the length of shortest prefix needed to get all blocks that occur.

A035326 Beginning of last prime pattern of length n to appear among positive integers.

Original entry on oeis.org

4, 8, 9, 24, 25, 90, 91, 114, 143, 143, 205, 205, 1601, 1601, 2327, 2327, 5467, 5467, 18043, 18043, 29381, 29381, 90001, 90001, 3153569, 3153569, 3153569, 3153569, 4647259, 4647259, 74266243, 74266243

Offset: 1

David W. Wilson

nonn

			a(7) = 91 since 91 starts the length-7 prime pattern "ccccccp" (six composites followed by one prime) and all other possible length-7 prime patterns start with a number smaller than 91.

Cf. A023192.

A094660 Number of permissible patterns of primes in a fixed interval of n consecutive integers.

Original entry on oeis.org

0, 1, 2, 4, 6, 9, 12, 18, 24, 34, 44, 58, 72, 100, 128, 169, 210, 267, 324, 429, 534, 694, 854, 1064, 1274, 1657, 2040, 2571, 3102, 3780, 4458, 5801, 7144, 9067, 10990, 13472, 15954, 20356, 24758, 30607, 36456, 44280, 52104, 66168, 80232, 98524, 116816, 140797, 164778

Offset: 0

Thomas J Engelsma (tom(AT)opertech.com), Jun 09 2004

Similar to A023192. (Here we ignore the empty pattern and start at 0.) These are called "admissible constellations" of primes. - Don Reble, Jun 12 2004

			a(5)=9 because primes can exist in interval as x.... .x... ..x.. ...x. ....x x.x.. .x.x. ..x.x or x...x

Similar work at Permissible Patterns.

Cf. A008407, A020497, A023189. Equals A023192 - 1.

a(n) = Sum_{k=1..floor((n+1)/2)} (n + 2 - 2*k)*A023189(k). - Jon E. Schoenfield, May 17 2024

a(42)-a(48) from Pontus von Brömssen, Aug 25 2025

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A023189 Conjecturally, number of infinitely recurring prime patterns of width 2n-1.

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A023190 Conjecturally, maximum number of primes in an infinitely-recurring prime pattern of width 2*n-1.

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A280417 Number of distinct length-n blocks (a.k.a. subword complexity) of the characteristic sequence of the prime numbers A010051.

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A035326 Beginning of last prime pattern of length n to appear among positive integers.

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A094660 Number of permissible patterns of primes in a fixed interval of n consecutive integers.

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