A055382 -id:A055382 - cp's OEIS frontend

A055380 Central prime p in the smallest (2n+1)-tuple of consecutive primes that are symmetric with respect to p.

5, 18731, 683783, 98303927, 60335249959, 1169769749219, 3945769040699039, 159067808851610657, 6919940122097246597

Offset: 1

Jud McCranie, Jun 23 2000

Least n-tuply balanced primes: primes which are averages of both their immediate neighbors, their second neighbors, their third neighbors, ... and their n-th neighbors.
a(9) <= 6919940122097246597. The solution was found by the BOINC project "SPT test project". - Natalia Makarova, Nov 25 2023
a(n) is the smallest number m such that A346399(m) = 2n + 1. - Ya-Ping Lu, May 12 2024

			In 5-tuple of consecutive primes (18713, 18719, 18731, 18743, 18749), the primes are symmetric w.r.t. its central prime 18731, since 18713+18749 = 18719+18743 = 2*18731, and this is the smallest such 5-tuple. Hence, a(2)=18731.
Alternatively, the symmetry can be seen from the differences between consecutive primes. For (18713, 18719, 18731, 18743, 18749), the differences are (6,12,12,6).

Stop@home, BOINC project to search all up to 2^64. [Dead link]
Symmetric Prime Tuples, SPT test project.

Cf. A001223, A055381, A055382, A006562, A051795, A081415, A096710, A346399.

Table[i = n + 2;
 While[x = Differences[Table[Prime[k + i], {k, -n, n}]];
x != Reverse[x], i++]; Prime[i], {n, 3}] (* Robert Price, Oct 12 2019 *)

a(n) = A151800^(n)(A175309(2n)), i.e., A151800 applied n times on A175309(2n). - Max Alekseyev, Jul 26 2014

a(6) from Donovan Johnson, Mar 09 2008
Definition corrected by Max Alekseyev, Jul 29 2014
a(7) from Dmitry Petukhov, added by Max Alekseyev, Nov 03 2014
a(8) from SPT project, added by Dmitry Petukhov, Apr 06 2017
a(9) from SPT project, added by Dmitry Petukhov, Mar 25 2025

A175309 a(n) = the smallest prime prime(k) such that prime(k+j) - prime(k+j-1) = prime(n+k+1-j) - prime(n+k-j) for all j with 1 <= j <= n.

Original entry on oeis.org

2, 3, 5, 18713, 5, 683747, 17, 98303867, 13, 60335249851, 137, 1169769749111, 8021749, 3945769040698829, 1071065111, 159067808851610411, 1613902553, 6919940122097246303, 1797595814863

Offset: 1

Leroy Quet, Mar 27 2010

From M. F. Hasler, Apr 02 2010: (Start)
Also: Start of the first sequence of n+1 consecutive primes symmetrically distributed w.r.t. their barycenter, e.g., [2,3], [3,5,7], [5,7,11,13], [18713, 18719, 18731, 18743, 18749]. With this definition, it would make sense to prefix the sequence with an initial term a(0)=2.
Sequence A081235 (or A055382, which is essentially the same) consists of every other term of this sequence. (End)
a(19) = 1797595814863, a(21) = 633925574060671, a(23) = 22930603692243271. - Tomáš Brada, May 25 2020

Andrew Howroyd, Table of n, a(n) for n = 1..19
BOINC project to search all up to 2^64
Symmetric Prime Tuples, SPT test project

Cf. A006562, A051795, A081235, A081415, A096710, A055382.

A175309[n_] := Module[{k},
   k = 1; While[! AllTrue[Range[n], Prime[k+#] - Prime[k+#-1] ==
        Prime[n+k+1-#] - Prime[n+k-#] &], k++]; Return[Prime[k]]];
Table[A175309[n], {n, 1, 7}]  (* Robert Price, Mar 27 2019 *)

a(n)={ my( last=vector(n++,i,prime(i)), m, i=Mod(n-2,n)); forprime(p=last[n],default(primelimit), m=last[1+lift(i+2)]+last[1+lift(i++)]=p; for( j=1,n\2, last[1+lift(i-j)]+last[1+lift(i+j+1)]==m || next(2)); return( last[1+lift(i+1)])) } \\ M. F. Hasler, Apr 02 2010

isok(p, n) = {my(k=primepi(p)); for (j=1, n, if (prime(k+j) - prime(k+j-1) != prime(n+k+1-j) - prime(n+k-j), return (0));); return (1);} \\ Michel Marcus, Apr 08 2017

a(2n-1) = A081235(n) (= A055382(n) for n>1). - M. F. Hasler, Apr 02 2010

Terms through a(12) were calculated by (in alphabetical order) Franklin T. Adams-Watters, Hans Havermann and D. S. McNeil
Minor edits by N. J. A. Sloane, Apr 02 2010
a(14) from Dmitry Petukhov, added by Max Alekseyev, Nov 03 2014
a(16) from BOINC project, added by Dmitry Petukhov, Apr 06 2017
a(18)-a(19) from SPT test project, added by Dmitry Petukhov, Mar 16 2025

A055381 Smallest composite k such that the n closest primes below and above k are symmetric about k.

Original entry on oeis.org

4, 9, 12, 30, 30, 165, 8021811, 1071065190, 1613902650, 1797595815015, 633925574060895, 22930603692243585, 5179852391836339140, 9648166508472058455

Offset: 1

Jud McCranie, Jun 23 2000

Center of the smallest 2n-tuple of consecutive odd primes with symmetrical gaps (cf. A055382).

			The three primes on each side of 12 (13, 17, 19 and 11, 7, 5) are symmetrical with respect to the gaps, so a(3) = 12.

Carlos Rivera, Problem 60. Symmetric primes on each side, The Prime Puzzles & Problems Connection.

Cf. A000040, A000720, A001223, A055380, A055382, A151800, A359440.

Table[i = n + 2;
 While[x =
   Differences@
    Flatten@{Table[NextPrime[i, k], {k, -n, -1}], i,
      Table[NextPrime[i, k], {k, 1, n}]}; x != Reverse[x],
i++]; i, {n, 6}] (* Robert Price, Oct 12 2019 *)

a(n) = ( A055382(n) + A000040(A000720(A055382(n))+2n) ) / 2 = ( A055382(n) + A151800(...(A151800(A055382(n)))...) ) / 2, where A151800 is iterated 2n times. - Max Alekseyev, Jul 23 2015

a(n) = (A000040(m) + A000040(m+1))/2, where m = min( {k >= 2 : A359440(k) >= n-1} ). - Peter Munn, Jan 09 2023

a(10) from Donovan Johnson, Mar 09 2008
a(11) from Dmitry Petukhov, added by Max Alekseyev, Aug 08 2014
a(12) computed from A055382(12) by Max Alekseyev, Jul 23 2015
Name clarified by Peter Munn, Jan 09 2023
a(13)-a(14) computed from A055382 by Dmitry Petukhov, Mar 25 2025

A081235 Smallest prime starting a sequence of 2n consecutive primes with symmetrical gaps about the center.

Original entry on oeis.org

2, 5, 5, 17, 13, 137, 8021749, 1071065111, 1613902553, 1797595814863, 633925574060671, 22930603692243271, 5179852391836338871, 9648166508472058129

Offset: 1

Christopher Hunt Gribble and T. D. Noe, Apr 02 2010

			The first term is 2 since the 2 primes 2, 3 have a gap of 1, which is trivially symmetric about its center.
The second term is 5 since the 4 primes 5, 7, 11, 13 have gaps 2, 4, 2, which is symmetric about its center.
The twelve primes 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193 have gaps 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2 - symmetric about the middle, so a(6) = 137.

N. Makarova and others, Distributed computing project, discussion at the scientific forum dxdy.ru (in Russian), Feb. 2015.
Symmetric Prime Tuples, SPT test project
Index entries for primes, gaps between

Cf. A000040, A001223, A055380, A055381, A175309, A359440.

A variant of A055382.

A081235(n) = { my(last=vector(n*=2,i,prime(i)), m, i=Mod(n-2,n)); forprime(p=last[n],default(primelimit), m=last[1+lift(i+2)]+last[1+lift(i++)]=p; for(j=1,n\2,last[1+lift(i-j)]+last[1+lift(i+j+1)]==m||next(2)); return(last[1+lift(i+1)]))} \\ M. F. Hasler, Apr 02 2010

a(n) = A175309(2n-1) (= A055382(n) for n>1). [M. F. Hasler, Apr 02 2010]

a(n) = A000040(k), where k = least number such that A359440(k+n-1) >= n-1. - Peter Munn, Jan 05 2023

a(11) from Dmitry Petukhov, added by Max Alekseyev, Aug 08 2014
a(12) from an anonymous participant of the project, added by Natalia Makarova, Jul 16 2015
a(13)-a(14) from SPT test project, added by Dmitry Petukhov, Mar 16 2025

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.

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.

A336967 Prime starting a sequence of 24 consecutive primes with symmetrical gaps about the center.

Original entry on oeis.org

22930603692243271, 34984922852185283, 60960572612579749, 226721453950385059, 301850075265898823, 310402815525745511, 341206644560627711, 357582484287837103, 481408770994035947, 492720459594614777, 528050771271601307, 587950582712698157, 675424273001524577

Offset: 1

Tomáš Brada, Aug 09 2020

nonn

Tomáš Brada, Table of n, a(n) for n = 1..18
Tomáš Brada, Natalia Makarova, Symmetric Prime Tuples project
Natalia Makarova, About Stop@home project
Natalia Makarova and Carlos Rivera, Problem 62. Symmetric k-tuples of consecutive primes, The Prime Puzzles and Problems Connection.

Cf. A000040, A055381, A055382, A064101, A081235, A175309, A335044, A335394, A336966, A336968, A359440.

Primes p = prime(k) = A000040(k) such that A359440(k+11) >= 11. - Peter Munn, Jan 09 2023

A336968 Prime starting a sequence of 22 consecutive primes with symmetrical gaps about the center.

Original entry on oeis.org

633925574060671, 2235053194261739, 3693434256575461, 6244996197964523, 7312449941282693, 11768508587048027, 12241378636561883, 12696156429346387, 13388148635660387, 14052415423668901, 18620445306703861, 19802687937976219, 22930603692243341, 23122811970297833

Offset: 1

Tomáš Brada, Aug 09 2020

nonn

Tomáš Brada, Table of n, a(n) for n = 1..345
Tomáš Brada, Natalia Makarova, Symmetric Prime Tuples project
Natalia Makarova, About Stop@home project
Natalia Makarova and Carlos Rivera, Problem 62. Symmetric k-tuples of consecutive primes, The Prime Puzzles and Problems Connection.

Cf. A000040, A055381, A055382, A064101, A081235, A175309, A333977, A335044, A335394, A336967, A359440.

Primes p = prime(k) = A000040(k) such that A359440(k+10) >= 10. - Peter Munn, Jan 09 2023

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

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

cp's OEIS Frontend

A055380 Central prime p in the smallest (2n+1)-tuple of consecutive primes that are symmetric with respect to p.

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A175309 a(n) = the smallest prime prime(k) such that prime(k+j) - prime(k+j-1) = prime(n+k+1-j) - prime(n+k-j) for all j with 1 <= j <= n.

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A055381 Smallest composite k such that the n closest primes below and above k are symmetric about k.

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A081235 Smallest prime starting a sequence of 2n consecutive primes with symmetrical gaps about the center.

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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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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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A336967 Prime starting a sequence of 24 consecutive primes with symmetrical gaps about the center.

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A336968 Prime starting a sequence of 22 consecutive primes with symmetrical gaps about the center.

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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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