A266512 -id:A266512 - cp's OEIS frontend

A266511 Minimal difference between the smallest and largest of n consecutive large primes that form a symmetric n-tuplet as permitted by divisibility considerations.

0, 2, 12, 8, 36, 16, 60, 26, 84, 34, 132, 46, 168, 56, 180, 74, 240, 82, 252, 94, 324, 106, 372, 118, 420, 134, 432, 142, 492, 146, 540, 158, 600, 166, 648, 178, 660, 194, 720, 202, 780, 214, 816, 226, 840, 254, 912, 262, 1020, 278

Offset: 1

Max Alekseyev, Dec 30 2015

For the definition of n-tuplet and minimal differences without the symmetry restriction, see A008407. In particular, a(n) >= A008407(n).
An n-tuplet (p(1),...,p(n)) is symmetric if p(k) + p(n+1-k) is the same for all k=1,2,...,n (cf. A175309).
Smallest primes starting a shortest symmetric n-tuplet are given in A266512.
For odd n, a(n) is divisible by 12.

			For n=3, any shortest symmetric n-tuplet has the form (p, p+6, p+12) and thus a(3)=12.
From _Jon E. Schoenfield_, Jan 05 2016: (Start)
For each n-tuplet (p(1), ..., p(n)) with odd n, let m be its middle prime, i.e., m = p((n+1)/2). Then, since (by symmetry) (p(k) + p(n+1-k))/2 = m for all k = 1..n, we can define the n-tuplet by m and its vector of differences d(j) = m - p(j) for j = 1..(n-1)/2. In other words, given m and d(j) for j = 1..(n-1)/2, the (n-1)/2 primes below m are given by p(j) = m - d(j), and the (n-1)/2 primes above m are given by p(n+1-j) = m + d(j); the difference p(n) - p(1) is thus (m + d(1)) - (m - d(1)) = 2*d(1).
For example, one symmetric 7-tuplet of consecutive primes is (12003179, 12003191, 12003197, 12003209, 12003221, 12003227, 12003239), which can be written as (m-30, m-18, m-12, m, m+12, m+18, m+30) where m=12003209; here we have d(1)=30, d(2)=18, d(3)=12. Among all symmetric 7-tuplets of consecutive primes that satisfy divisibility considerations, the minimal value of d(1) is, in fact, 30, so a(7) = 2*30 = 60.
For n = 3, 5, ..., 29, the lexicographically first vector (d(1), d(2), ..., d((n-1)/2)) permitted by divisibility considerations is as follows:
   n|  1   2   3   4   5   6   7   8   9  10  11  12  13  14
  --+-------------------------------------------------------
   3|  6
   5| 18  12
   7| 30  18  12
   9| 42  30  18  12
  11| 66  60  36  24   6
  13| 84  66  60  36  24   6
  15| 90  84  66  60  36  24   6
  17|120 108  90  78  60  48  42  18
  19|126 120 114  96  84  54  36  30   6
  21|162 150 132 120 108 102  78  48  42  18
  23|186 180 150 144 126  96  84  66  60  54  30
  25|210 186 180 150 144 126  96  84  66  60  54  30
  27|216 210 204 180 126 120 114  96  84  54  36  30   6
  29|246 216 210 204 186 174 144 126  90  84  66  60  24   6
(End)

N. Makarova and C. Rivera, Problem 62. Symmetric k-tuples of consecutive primes.

Cf. A008407, A175309, A266512, A266676.

a(1)-a(10) from Natalia Makarova
a(11)-a(14), a(16) from Dmitry Petukhov
a(15) and a(17)-a(18) from Jaroslaw Wroblewski
a(20) from Natalia Makarova and Jaroslaw Wroblewski
a(19), a(21), a(23), a(25), a(27), a(29) from Jon E. Schoenfield, Jan 02 2016, Jan 05 2016
a(22), a(24), a(26), a(28), a(30) from Natalia Makarova, Jul 06 2016
a(31)-a(50) from Vladimir Chirkov, Jul 08 2016

A335044 Primes starting 14-tuples of consecutive primes that have symmetrical gaps about their mean and form 7 pairs of twin primes.

Original entry on oeis.org

1855418882807417, 2485390773085247, 4038284355308309, 14953912258447817, 16152884167551797, 20149877129714999, 23535061700758967, 24067519779525107, 25892136591156917, 28681238268465371, 29359755788438639, 38364690814563809, 52367733685120277

Offset: 1

Tomáš Brada, Jun 05 2020

nonn

			a(1) = A274792(7) = 1855418882807417 starts a 14-tuple of consecutive primes: 1855418882807417+s for s in {0 2 12 14 30 32 72 74 114 116 132 134 144 146} that are symmetric about 1855418882807417+73 and form 7 pairs of twin primes.

Tomáš Brada, Table of n, a(n) for n = 1..122

Cf. A330278, A274792, A055380, A055381, A055382, A081235, A266511, A266512.

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

A335394 Primes starting 16-tuples of consecutive primes that have symmetrical gaps about their mean and form 8 pairs of twin primes.

Original entry on oeis.org

2640138520272677, 119890755200639999, 156961225134536189, 193609877401516181, 215315384130681929, 404072710417411769, 517426190585100089, 519460320704755811

Offset: 1

Tomáš Brada, Natalia Makarova Jun 05 2020

			a(1) = A274792(8) = 2640138520272677 starts a 16-tuple of consecutive primes: 2640138520272677+s for s in {0, 2, 12, 14, 30, 32, 54, 56, 90, 92, 114, 116, 132, 134, 144, 146} that are symmetric about 2640138520272677+73 and form 8 pairs of twin primes.

Cf. A330278, A274792, A055380, A055381, A055382, A081235, A266511, A266512.

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.

A330278 Primes starting 12-tuples of consecutive primes that have symmetrical gaps about their mean and form 6 pairs of twin primes.

Original entry on oeis.org

17479880417, 158074620437, 1071796554401, 1087779101699, 1153782400787, 1628444511389, 2066102452949, 2083857437327, 2561560206377, 3731086236287, 3751571181929, 4158362831639, 4878193583477, 5008751356547, 5378606656847, 5531533689527, 7020090738707, 7036216236989

Offset: 1

Max Alekseyev, Dec 08 2019

nonn

			a(1) = A274792(6) = 17479880417 starts a 12-tuple of consecutive primes: 17479880417+s for s in {0, 2, 24, 26, 30, 32, 54, 56, 60, 62, 84, 86} that are symmetric about 17479880417+43 and form 6 pairs of twin primes.

Tomáš Brada, Table of n, a(n) for n = 1..10000 (terms a(7)-a(132) from Giovanni Resta, terms a(133)-a(228) from Franz-Xaver Harvanek)

Cf. A055380, A055381, A055382, A081235, A266511, A266512.

a(2)-a(6) from Franz-Xaver Harvanek

More terms from Giovanni Resta, Dec 10 2019

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

A336966 Primes starting 10-tuples of consecutive primes that have symmetrical gaps about their mean and form 5 pairs of twin primes.

Original entry on oeis.org

3031329797, 5188151387, 14168924459, 14768184029, 18028534367, 26697800819, 26919220961, 29205326387, 32544026699, 39713433671, 45898528799, 48263504459, 50791655009, 66419473031, 71525244611, 80179195037, 83700877199, 86767580069, 97660776137, 108116163479

Offset: 1

Tomáš Brada, Aug 09 2020

nonn

			a(1) = A274792(5) = 3031329797 starts a 10-tuple of consecutive primes: 3031329797+s for s in {0, 2, 12, 14, 42, 44, 72, 74, 84, 86} that are symmetric about 3031329797+43 and form 5 pairs of twin primes.

Cf. A055380, A055381, A055382, A081235, A266511, A266512, A335044, A335394, A336966, A336968.

cp's OEIS Frontend

A266511 Minimal difference between the smallest and largest of n consecutive large primes that form a symmetric n-tuplet as permitted by divisibility considerations.

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A335044 Primes starting 14-tuples of consecutive primes that have symmetrical gaps about their mean and form 7 pairs of twin primes.

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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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A335394 Primes starting 16-tuples of consecutive primes that have symmetrical gaps about their mean and form 8 pairs of twin primes.

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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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A330278 Primes starting 12-tuples of consecutive primes that have symmetrical gaps about their mean and form 6 pairs of twin 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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A336966 Primes starting 10-tuples of consecutive primes that have symmetrical gaps about their mean and form 5 pairs of twin primes.

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