A055380 -id:A055380 - cp's OEIS frontend

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

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

A269043 a(n) is the number of distinct values that can be expressed as prime(n+k) + prime(n-k) in at least 2 different ways.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 2, 3, 3, 2, 2, 3, 1, 4, 4, 2, 4, 4, 4, 3, 5, 5, 7, 9, 8, 7, 8, 7, 6, 7, 9, 7, 9, 8, 11, 8, 8, 7, 10, 9, 11, 12, 9, 9, 14, 11, 12, 11, 15, 15, 12, 14, 12, 12, 17, 11, 14, 15, 15, 14, 15, 18, 16, 13, 18, 12, 16, 14, 16, 14, 12, 19, 17, 13, 19

Offset: 1

Michel Lagneau, Feb 18 2016

nonn

Conjecture: a(n) > 0 for n > 3.

			a(13) = 3 because:
p(13 + 1)  + p(13 - 1)  = 43 + 37 = 80;
p(13 + 2)  + p(13 - 2)  = 47 + 31 = 78;
p(13 + 3)  + p(13 - 3)  = 53 + 29 = 82;
p(13 + 4)  + p(13 - 4)  = 59 + 23 = 82;
p(13 + 5)  + p(13 - 5)  = 61 + 19 = 80;
p(13 + 6)  + p(13 - 6)  = 67 + 17 = 84;
p(13 + 7)  + p(13 - 7)  = 71 + 13 = 84;
p(13 + 8)  + p(13 - 8)  = 73 + 11 = 84.
p(13 + 9)  + p(13 - 9)  = 79 + 7  = 86;
p(13 + 10) + p(13 - 10) = 83 + 5  = 88;
p(13 + 11) + p(13 - 11) = 89 + 3  = 92;
p(13 + 12) + p(13 - 12) = 97 + 2  = 99.
The 3 distinct values of prime(n+k) + prime(n-k) that are each obtained in at least 2 ways are 80, 82 and 84.

Michel Lagneau, Table of n, a(n) for n = 1..1000

Cf. A006562, A055380, A055382.

for n from 1 to 100 do:
  lst:={}:W:=array(1..n-1):cr:=0:
    for m from n-1 by -1 to 1 do:
      q:=ithprime(n-m)+ithprime(n+m):lst:=lst union {q}:W[m]:=q:
    od:
      n0:=nops(lst):c:=0:U:=array(1..n0):
        for i from 1 to n0 do:
         c1:=0:
         for j from 1 to n-1 do:
          if lst[i]=W[j] then c:=c+1:c1:=c1+1:
          else fi:
         od:
        U[i]:=c1:cr:=cr+1:
       od:
       ct:=0:
       for l from 1 to cr do:
       if U[l]>1 then ct:=ct+1:
       else fi:
       od:
       printf(`%d, `,ct):
od:

a(n) = {v = []; for (k=1, n-1, v = concat(v, prime(n+k) + prime(n-k));); vd = vecsort(v,,8); sum(k=1, #vd, #select(x->x==vd[k], v)>1);} \\ Michel Marcus, Mar 13 2016

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.

A122120 a(n) = 4a(n-1) + 9a(n-2), for n>1, with a(0)=1, a(1)=3.

Original entry on oeis.org

1, 3, 21, 111, 633, 3531, 19821, 111063, 622641, 3490131, 19564293, 109668351, 614752041, 3446023323, 19316861661, 108281656551, 606978381153, 3402448433571, 19072599164661, 106912432560783, 599303122725081

Offset: 0

Philippe Deléham, Oct 19 2006

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,9).

First differences of A015533.

Binomial transform of A091914.

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)/(1-4*x-9*x^2) )); // G. C. Greubel, Feb 26 2019

CoefficientList[Series[(1-x)/(1-4*x-9*x^2), {x, 0, 30}], x] (* G. C. Greubel, Feb 26 2019 *)
nxt[{a_,b_}]:={b,4b+9a}; NestList[nxt,{1,3},20][[All,1]] (* or *) LinearRecurrence[{4,9},{1,3},30] (* Harvey P. Dale, Oct 06 2020 *)

my(x='x+O('x^30)); Vec((1-x)/(1-4*x-9*x^2)) \\ G. C. Greubel, Feb 26 2019

((1-x)/(1-4*x-9*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 26 2019

a(n) = Sum_{k=0..n} 3^(n-k)*A055380(n,k).
G.f.: (1-x)/(1-4*x-9*x^2).
Limit_{n -> oo} a(n+1)/a(n) = 2 + sqrt(13).

cp's OEIS Frontend

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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A269043 a(n) is the number of distinct values that can be expressed as prime(n+k) + prime(n-k) in at least 2 different ways.

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Maple

PARI

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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Author

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Examples

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A122120 a(n) = 4*a(n-1) + 9*a(n-2), for n>1, with a(0)=1, a(1)=3.

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PARI

A122120 a(n) = 4a(n-1) + 9a(n-2), for n>1, with a(0)=1, a(1)=3.