A023189 -id:A023189 - cp's OEIS frontend

A023192 Conjecturally, number of infinitely-recurring prime patterns on n consecutive integers.

2, 3, 5, 7, 10, 13, 19, 25, 35, 45, 59, 73, 101, 129, 170, 211, 268, 325, 430, 535, 695, 855, 1065, 1275, 1658, 2041, 2572, 3103, 3781, 4459, 5802, 7145, 9068, 10991, 13473, 15955, 20357, 24759, 30608, 36457, 44281, 52105, 66169, 80233, 98525, 116817, 140798, 164779

Offset: 1

David W. Wilson

			a(3) = 5: Conjecturally, there are five infinitely-recurring prime patterns of length 3. These are "ccc" (three composites in a row), "ccp", "cpc", "pcc" and "pcp". Others, like "ppc", starting at 2, only occur a finite number of times.

Pontus von Brömssen, Table of n, a(n) for n = 1..120 (terms 1..76 from Sean A. Irvine)
Sean A. Irvine, Java program (github)

Cf. A023189, A035326.

a(n) = 1 + Sum_{k=1..floor((n+1)/2)} (n-2*k+2)*A023189(k). - Pontus von Brömssen, Aug 25 2025

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

A292225 Row sums of irregular triangle A292224. a(n) gives the total number of admissible tuples starting with 0 in the interval [0, 1, ..., n-1].

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 6, 6, 10, 10, 14, 14, 28, 28, 41, 41, 57, 57, 105, 105, 160, 160, 210, 210, 383, 383, 531, 531, 678, 678, 1343, 1343, 1923, 1923, 2482, 2482, 4402, 4402, 5849, 5849, 7824, 7824, 14064, 14064, 18292, 18292, 23981, 23981, 39745, 39745, 57307, 57307, 71639, 71639, 117846, 117846

Offset: 1

Wolfdieter Lang, Oct 09 2017

nonn

This sequence is given in column 2 of Table 2, p. 27, of the Engelsma link.

See A292224 for the reason for the repetitions for n = 2*k+1 and n = 2*(k+1) for k >= 0, the definition of "admissible", references, and examples of these admissible k-tuples for n = 1..10 (with k = 1, 2, ..., A023193(n)).

Thomas J. Engelsma, Permissible Patterns of Primes, September 2009, Table 2, p. 27.

Cf. A023189, A023193, A292224.

a(n) = Sum_{k=1..A023193(n)} A292224(n, k), for n >= 1.

a(2*n+1) = a(2*n) + A023189(n+1). - Pontus von Brömssen, Aug 21 2025

Terms a(27) .. a(56) from Engelsma's Table 2 (there are also a(57)..a(62) given but a(62) should be 364545 if a(61) = 364545 is correct). - Wolfdieter Lang, Oct 17 2017

cp's OEIS Frontend

A023192 Conjecturally, number of infinitely-recurring prime patterns on n consecutive integers.

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

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A292225 Row sums of irregular triangle A292224. a(n) gives the total number of admissible tuples starting with 0 in the interval [0, 1, ..., n-1].

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