A261324 -id:A261324 - cp's OEIS frontend

A065688 First prime in the smallest (nontrivial) prime k-tuplet.

2, 3, 5, 5, 5, 7, 11, 11, 11, 11, 11, 11, 11, 11, 11, 13, 13, 13, 13, 29, 29, 19, 19

Offset: 1

Frank Ellermann, Dec 04 2001

For a proper definition see the cross-references.

			a(8): 11 13 17 19 23 29 31 37 are primes and 37-11=26=A008407(8).

T. Forbes, Norman Luhn, Prime k-tuplets
Eric Weisstein's World of Mathematics, k-Tuple Conjecture

Cf. A008407 (minimal difference of first and last prime in a prime k-tuplet), A023193 (prime k-tuplet conjectures), A047947 (Schinzel's rhobar), A020497.

Cf. A261324 (another variant including trivial tuplets).

a(1) prepended and a(20)-a(23) added by Max Alekseyev, Aug 15 2015

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

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A065688 First prime in the smallest (nontrivial) prime k-tuplet.

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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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