A065688 -id:A065688 - cp's OEIS frontend

A065706 Least member p1 of prime octuplets (p1, p2, p3, ..., p8 = p1 + 26), the eight p's being consecutive primes.

11, 17, 1277, 88793, 113147, 284723, 855713, 1146773, 2580647, 6560993, 15760091, 20737877, 25658441, 58208387, 69156533, 73373537, 74266253, 76170527, 93625991, 100658627, 134764997, 137943347, 165531257, 171958667

Offset: 1

Frank Ellermann, Dec 05 2001

nonn

3 patterns for 8-tuplets: 11010011001011, 11011010011001 and v.v.

See A022011, A022012 and A022013 for the three different possible patterns. The sequence is conjectured to be infinite, although it is not even proved that there are infinitely many twin primes (p1, p2 = p1+2). - M. F. Hasler, May 02 2015

			a(3) = 1277, 1279, 1283, 1289, 1291, 1297, 1301, 1303 = 1277+26 are primes.

Harry J. Smith and Dana Jacobsen, Table of n, a(n) for n = 1..18123 [first 100 terms from Harry J. Smith]
Tony Forbes and Norman Luhn, Prime k-tuplets
Norman Luhn, The smallest prime k-tuplets, database of compressed files.
Eric Weisstein's World of Mathematics, k-Tuple Conjecture

11 = A065688(8), 26 = A008407(8), 8 = A023193(26+1), octets in A066082 are another (not minimal) constellation of 8 primes.
Union of A022011, A022012 and A022013.
See A257124 (prime septuplets) with an overview of prime k-tuplets.

{ n=0; p1=2; p8=19; for (m=1, 10^12, p1=nextprime(p1+1); p8=nextprime(p8+1); if (p8 - p1 == 26, write("b065706.txt", n++, " ", p1); if (n==100, return)) ) } \\ Harry J. Smith, Oct 26 2009

use ntheory ":all"; my($s,$e,$i,%h)=(1,1e10,0); undef @h{sieve_prime_cluster($s,$e,2,6,8,12,18,20,26), sieve_prime_cluster($s,$e,2,6,12,14,20,24,26), sieve_prime_cluster($s,$e,6,8,14,18,20,24,26)}; say ++$i," $" for sort {$a<=>$b} keys %h; # _Dana Jacobsen, Oct 10 2015

A266512 Smallest prime starting a (nonsingular) symmetric n-tuplet of the shortest span (=A266511(n)).

Original entry on oeis.org

2, 3, 47, 5, 18713, 7, 12003179, 17, 1480028129, 13, 1542186111157, 41280160361347, 660287401247633, 10421030292115097, 3112462738414697093, 996689250471604163, 258406392900394343851, 824871967574850703732309, 9425346484752129657862217, 824871967574850703732303

Offset: 1

Max Alekseyev, Dec 30 2015

A similar sequence that allows singular symmetric n-tuples is given in A266583.
a(1)-a(10) from Natalia Makarova
a(11)-a(14), a(16) from Dmitry Petukhov
a(15), a(17) from Jaroslaw Wroblewski

N. Makarova and Carlos Rivera, Problem 62. Symmetric k-tuples of consecutive primes, The Prime Puzzles & Problems Connection.
Natalia Makarova and Vladimir Chirkov, Theoretical patterns with a minimal diameter for a(2) - a(50)
Dmitry Petukhov, Anton Nikonov and Ruslan Vikulov, found a(19) and proof of that is smallest (in Russian), posts in Symmetric tuples of consecutive primes on dxdy blog.

Cf. A055380, A065688, A175309, A266511, A266583, A266584.

a(n) = A000040(A266584(n)).

a(18) from Jaroslaw Wroblewski
a(20) from Natalia Makarova and Jaroslaw Wroblewski
a(19) from Dmitry Petukhov, Anton Nikonov and Ruslan Vikulov, Jan 24 2025

A083409 Number of prime k-tuplet constellations, i.e., patterns with minimal diameter A008407.

Original entry on oeis.org

1, 2, 1, 2, 1, 2, 3, 4, 2, 2, 2, 6, 2, 4, 2, 4, 2, 4, 2, 2, 4, 2, 4, 18, 2, 8, 10, 2, 2, 2, 4, 14, 20, 2, 2, 2, 6, 26, 26, 8, 2, 6, 18, 4, 4, 4, 2, 2, 22, 22, 2, 2, 26, 6, 6, 2, 2, 4, 2, 2, 6, 2, 2, 2, 2, 18, 2, 20, 2, 2, 2, 10, 2, 14, 14, 40, 8, 2, 14, 14, 16, 4, 2, 2, 60, 50, 2, 2, 2, 16, 2, 18, 12

Offset: 2

Frank Ellermann, Jun 07 2003

nonn

			For a(8) = 3 octuplet patterns see A065706. for a(6) = 1 sextet see A061671.

T. D. Noe, Table of n, a(n) for n=2..305 (from Engelsma's data)
Thomas J Engelsma, Permissible Patterns.
Tony Forbes and Norman Luhn, Patterns of prime k-tuplets & the Hardy-Littlewood constants.
Eric Weisstein's World of Mathematics, k-Tuple Conjecture.

Cf. A008407, A020497, A023193, A061671, A065688, A065706.

More terms from Engelsma's website sent by T. D. Noe, Jul 21 2006

A261324 Smallest prime p that starts an n-tuplet of consecutive primes of length A008407(n).

Original entry on oeis.org

2, 3, 5, 3, 5, 7, 11, 11, 7, 5, 5, 11, 11, 11, 11, 7, 13, 13, 13, 29, 29, 7, 7

Offset: 1

Max Alekseyev, Aug 14 2015

In contrast to A065688, n-tuplets here may be singular and give the complete set of residues modulo some prime. For example, for n=4 we have the 4-tuplet: (3,5,7,11) = (3,3+2,3+4,3+8), but there are no other prime 4-tuplets of the form (p,p+2,p+4,p+8), since one of its elements would be divisible by 3.

For any n, a(n) <= n or a(n) = A065688(n).

Cf. A000040, A008407, A065688, A261323.

a(n) = A000040(A261323(n)).

A261323 Smallest m such that prime(m+n-1) - prime(m) = A008407(n); that is, prime(m) starts the smallest n-tuplet of consecutive primes of length A008407(n).

Original entry on oeis.org

1, 2, 3, 2, 3, 4, 5, 5, 4, 3, 3, 5, 5, 5, 5, 4, 6, 6, 6, 10, 10, 4, 4

Offset: 1

Max Alekseyev, Aug 14 2015

See A261324 for further comments and the relation to A065688.

Cf. A008407, A261324.

A261323(n,d=A008407[n],m=0)={until(prime(m+n)==prime(m++)+d,);m} \\ Assumes a precomputed vector A008407 with at least n elements, or supply the gap as 2nd arg. Inefficient for n>23. - M. F. Hasler, Aug 17 2015

a(n) = A000720(A261324(n)). - M. F. Hasler, Aug 17 2015

A266583 Smallest prime starting a symmetric n-tuple of consecutive primes of the smallest span (=A266676(n)).

Original entry on oeis.org

2, 2, 3, 5, 18713, 5, 12003179, 17, 1480028129, 13, 1542186111157, 41280160361347, 660287401247633, 10421030292115097, 3112462738414697093, 996689250471604163, 258406392900394343851, 824871967574850703732309, 9425346484752129657862217, 824871967574850703732303

Offset: 1

Max Alekseyev, Jan 01 2016

An n-tuple (p(1),...,p(n)) is symmetric if p(k)+p(n+1-k) is the same for all k=1,2,...,n (cf. A175309).
In contrast to A266512, n-tuples here may be singular and give the complete set of residues modulo some prime. For example, for n=3 we have the symmetric 3-tuple: (3,5,7) = (3,3+2,3+4), but there are no other symmetric 3-tuples of the form (p,p+2,p+4), since one of its elements would be divisible by 3.
For any n, a(n) <= n or a(n) = A266512(n).

Dmitry Petukhov, Anton Nikonov and Ruslan Vikulov, found a(19) and proof of that is smallest (in Russian), posts in Symmetric tuples of consecutive primes on dxdy forum.

Cf. A055380, A065688, A175309, A266511, A266512, A261324.

a(n) = A000040(A266585(n)).

a(18)-a(20) added by Dmitry Petukhov, Feb 15 2025

A266585 Smallest m such that prime(m) starts a symmetric n-tuple of consecutive primes of the smallest span (=A266676(n)).

Original entry on oeis.org

1, 1, 2, 3, 2136, 3, 788244, 7, 73780392, 6, 57067140928, 1361665032086, 19953429852608, 290660101635794, 74896929428416952, 24660071077535201, 5620182896687887031

Offset: 1

Max Alekseyev, Jan 01 2016

See A266583 for further comments and the relation to A266584.

A000040(a(n)+n-1) - A000040(a(n)) = A266676(n).

Cf. A055380, A065688, A175309, A266511, A266512, A261323, A261324, A266583, A266584.

a(n) = A000720(A266583(n)).

More terms from Max Alekseyev, Jul 24 2019

A266676 Smallest span (difference between the start and end) of a symmetric n-tuple of consecutive primes.

Original entry on oeis.org

0, 1, 4, 8, 36, 14, 60, 26, 84, 34, 132, 46, 168, 56, 180, 74, 240, 82

Offset: 1

Max Alekseyev, Jan 02 2016

An n-tuple (p(1),...,p(n)) is symmetric if p(k)+p(n+1-k) is the same for all k=1,2,...,n (cf. A175309).
In contrast to A266511, n-tuples here may be singular and give the complete set of residues modulo some prime. For example, for n=3 we have the symmetric 3-tuple: (3,5,7) = (3,3+2,3+4) of span a(3)=4, but there are no other symmetric 3-tuples of the form (p,p+2,p+4), since one of its elements would be divisible by 3.
a(n) <= A266511(n).

The smallest starting primes and their indices of the corresponding tuples are given in A266583 and A266585.

Cf. A055380, A065688, A175309, A266511, A266512, A261324.

A266584 Smallest m such that prime(m) starts a (nonsingular) symmetric n-tuplet of consecutive primes of the smallest span (=A266511(n)).

Original entry on oeis.org

1, 2, 15, 3, 2136, 4, 788244, 7, 73780392, 6, 57067140928, 1361665032086, 19953429852608, 290660101635794, 74896929428416952, 24660071077535201, 5620182896687887031

Offset: 1

Max Alekseyev, Jan 01 2016

See A266583 for further comments and the relation to A266585.

A000040(a(n)+n-1) - A000040(a(n)) = A266511(n).

Cf. A055380, A065688, A175309, A266511, A266512, A261323, A261324, A266583, A266585.

a(n) = A000720(A266512(n)).

More terms from Max Alekseyev, Jul 24 2019

cp's OEIS Frontend

A065706 Least member p1 of prime octuplets (p1, p2, p3, ..., p8 = p1 + 26), the eight p's being consecutive primes.

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A266512 Smallest prime starting a (nonsingular) symmetric n-tuplet of the shortest span (=A266511(n)).

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A083409 Number of prime k-tuplet constellations, i.e., patterns with minimal diameter A008407.

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A261324 Smallest prime p that starts an n-tuplet of consecutive primes of length A008407(n).

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A261323 Smallest m such that prime(m+n-1) - prime(m) = A008407(n); that is, prime(m) starts the smallest n-tuplet of consecutive primes of length A008407(n).

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A266583 Smallest prime starting a symmetric n-tuple of consecutive primes of the smallest span (=A266676(n)).

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A266585 Smallest m such that prime(m) starts a symmetric n-tuple of consecutive primes of the smallest span (=A266676(n)).

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A266676 Smallest span (difference between the start and end) of a symmetric n-tuple of consecutive primes.

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A266584 Smallest m such that prime(m) starts a (nonsingular) symmetric n-tuplet of consecutive primes of the smallest span (=A266511(n)).

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