A081235 -id:A081235 - cp's OEIS frontend

A055382 Smallest prime starting a sequence of 2n consecutive odd primes with symmetrical gaps about the center.

3, 5, 5, 17, 13, 137, 8021749, 1071065111, 1613902553, 1797595814863, 633925574060671, 22930603692243271, 5179852391836338871, 9648166508472058129

Offset: 1

Jud McCranie, Jun 23 2000

a(13) <= 5179852391836338871. The solution was found by the BOINC project "SPT test project". - Natalia Makarova, Dec 06 2023

			The first term is 3 since the 2 primes 3, 5 have a gap of 2, 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.

Discussion at the scientific forum dxdy.ru, Distributed computing project (in Russian)
Symmetric Prime Tuples, SPT test project
Index entries for primes, gaps between

Cf. A001223, A055380, A055381, A175309.

See A081235 for another version.

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

For n>1, a(n) = A081235(n) = A175309(2n-1).

a(10) from Donovan Johnson, Mar 09 2008
Minor edits by N. J. A. Sloane, Apr 02 2010
a(11) from Dmitry Petukhov, added by Max Alekseyev, Aug 08 2014
a(12) from an anonymous participant of the project, added by Max Alekseyev, Jul 21 2015
a(13)-a(14) from SPT test project, added by Dmitry Petukhov, Mar 16 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

A359440 A measure of the extent of reflective symmetry in the pattern of primes around each prime gap: a(n) is the largest k such that prime(n-j) + prime(n+1+j) has the same value for each j in 0..k.

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 1, 0, 0, 4, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 2, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 5, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0

Offset: 1

Alexandre Herrera, Jan 01 2023

nonn

If the prime gaps above and below a prime p have the same length, p is called a balanced prime (see A006562). Likewise, if the prime gaps above and below the n-th prime gap have the same length, this gap might be called a balanced prime gap. These gaps correspond to nonzero terms a(n). Similarly, if a(n) >= 2, the n-th prime gap is the equivalent of a doubly balanced prime (A051795), and so on. - Peter Munn, Jan 08 2023

			For n = 1, prime(1) + prime(2) = 2 + 3 = 5; "prime(0)" does not exist, so a(1) = 0.
For n = 4:
  j = 0:  prime(4) + prime(5) =  7 + 11 = 18;
  j = 1:  prime(3) + prime(6) =  5 + 13 = 18;
  j = 2:  prime(2) + prime(7) =  3 + 17 = 20 != 18, so a(4) = 1.
For n = 5:
  j = 0:  prime(5) + prime(6) = 11 + 13 = 24;
  j = 1:  prime(4) + prime(7) =  7 + 17 = 24;
  j = 2:  prime(3) + prime(8) =  5 + 19 = 24;
  j = 3:  prime(2) + prime(9) =  3 + 23 = 26 != 24, so a(5) = 2.

Cf. A000040, A006562, A051795, A055381, A081235.

import sympy
offset = 1
N = 100
l = []
for n in range(offset,N+1):
    j = 0
    first_sum = sympy.prime(n-j)+sympy.prime(n+j+1)
    while (n-j) > 1:
        j += 1
        sum = sympy.prime(n-j)+sympy.prime(n+j+1)
        if sum != first_sum:
            break
    l.append(max(0,j-1))
print(l)

a(n) = min( {n-1} U {k : 0 <= k <= n-2 and prime(n-k-1) + prime(n+k+2) <> prime(n) + prime(n+1)} ). - Peter Munn, Jan 08 2023

Introductory phrase added to name by Peter Munn, Jan 08 2023

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

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.

A333977 Prime starting a sequence of 20 consecutive primes with symmetrical gaps about the center.

Original entry on oeis.org

1797595814863, 2375065608481, 4465545586753, 21818228348093, 67696772430071, 82116093014611, 155947272322087, 161980267642951, 169560139541641, 202619277419161, 285719200081877, 299828814652799, 314942862282899, 365706921997577

Offset: 1

Tomáš Brada, Sep 20 2020

nonn

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, A336967, A336968, A359440.

Primes p = prime(k) = A000040(k) such that A359440(k+9) >= 9. - Peter Munn, Jan 09 2023

cp's OEIS Frontend

A055382 Smallest prime starting a sequence of 2n consecutive odd primes with symmetrical gaps about the center.

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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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A359440 A measure of the extent of reflective symmetry in the pattern of primes around each prime gap: a(n) is the largest k such that prime(n-j) + prime(n+1+j) has the same value for each j in 0..k.

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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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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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Keywords

Examples

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A333977 Prime starting a sequence of 20 consecutive primes with symmetrical gaps about the center.

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