A359440 -id:A359440 - cp's OEIS frontend

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

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

A064101 Primes p = prime(k) such that prime(k) + prime(k+5) = prime(k+1) + prime(k+4) = prime(k+2) + prime(k+3).

Original entry on oeis.org

5, 7, 19, 31, 97, 131, 151, 293, 587, 683, 811, 839, 857, 907, 1013, 1097, 1279, 2347, 2677, 2833, 3011, 3329, 4217, 4219, 5441, 5839, 5849, 6113, 8233, 8273, 8963, 9433, 10301, 10427, 10859, 11953, 13513, 13597, 13721, 13931, 14713, 15629, 16057

Offset: 1

Robert G. Wilson v, Sep 17 2001

			The first ten primes are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29. Take just the fourth through the ninth and rearrange them such that the first pairs with the sixth, the second with the fifth and the third with the fourth as follows: 7 and 23, 11 and 19 and 13 and 17. All three pairs sum to 30. Therefore a(2) = 7.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000040, A022885, A359440.

A := {}: for n to 1000 do p1 := ithprime(n); p2 := ithprime(n+1); p3 := ithprime(n+2); p4 := ithprime(n+3); p5 := ithprime(n+4); p6 := ithprime(n+5); if `and`(p1+p6 = p2+p5, p2+p5 = p3+p4) then A := `union`(A, {p1}) end if end do; A := A;

a = {0, 0, 0, 0, 0, 0}; Do[ a = Delete[ a, 1 ]; a = Append[ a, Prime[ n ] ]; If[ a[ [ 1 ] ] + a[ [ 6 ] ] == a[ [ 2 ] ] + a[ [ 5 ] ] == a[ [ 3 ] ] + a[ [ 4 ] ], Print[ a[ [ 1 ] ] ] ], {n, 1, 20000} ] (* RGWv *)
Prime[Select[Range[100], Prime[#] + Prime[# + 5] == Prime[# + 1] + Prime[# + 4] && Prime[#] + Prime[# + 5] == Prime[# + 2] + Prime[# + 3] &]]
Select[Partition[Prime[Range[2000]],6,1],#[[1]]+#[[6]]==#[[2]]+#[[5]] == #[[3]]+ #[[4]]&][[All,1]] (* Harvey P. Dale, Jan 16 2022 *)

{ n=0; default(primelimit, 1500000); for (k=1, 10^9, p1=prime(k) + prime(k + 5); p2=prime(k + 1) + prime(k + 4); p3=prime(k + 2) + prime(k + 3); if (p1==p2 && p2==p3, write("b064101.txt", n++, " ", prime(k)); if (n==1000, break)) ) } \\ Harry J. Smith, Sep 07 2009

Primes p = prime(k) = A000040(k) such that A359440(k+2) >= 2. - Peter Munn, Jan 09 2023

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

A064102 Primes p = prime(k) such that prime(k) + prime(k+7) = prime(k+1) + prime(k+6) = prime(k+2) + prime(k+5) = prime(k+3) + prime(k+4).

Original entry on oeis.org

17, 149, 677, 853, 1277, 5437, 6101, 13499, 13921, 19853, 22073, 41863, 49667, 51307, 51797, 55799, 61637, 66337, 83227, 91121, 100957, 103963, 109111, 113147, 128747, 136309, 137933, 148157, 158849, 163117, 167249, 179033, 205171, 208927

Offset: 1

Robert G. Wilson v, Sep 17 2001

			17 + 43 = 19 + 41 = 23 + 37 = 29 + 31.

Harry J. Smith, Table of n, a(n) for n = 1..400

Cf. A000040, A022885, A064101, A359440.

a = {0, 0, 0, 0, 0, 0, 0, 0}; Do[ a = Delete[ a, 1 ]; a = Append[ a, Prime[ n ] ]; If[ a[ [ 1 ] ] + a[ [ 8 ] ] == a[ [ 2 ] ] + a[ [ 7 ] ] == a[ [ 3 ] ] + a[ [ 6 ] ] == a[ [ 4 ] ] + a[ [ 5 ] ], Print[ a[ [ 1 ] ] ] ], {n, 1, 10^4} ]
Select[Partition[Prime[Range[20000]],8,1],#[[1]]+#[[8]]==#[[2]]+#[[7]]==#[[3]]+#[[6]]==#[[4]]+#[[5]]&][[;;,1]] (* Harvey P. Dale, Jul 03 2025 *)

{ n=0; default(primelimit, 8300000); for (k=1, 10^9, p1=prime(k) + prime(k + 7); p2=prime(k + 1) + prime(k + 6); p3=prime(k + 2) + prime(k + 5); p4=prime(k + 3) + prime(k + 4); if (p1==p2 && p2==p3 && p3==p4, write("b064102.txt", n++, " ", prime(k)); if (n==400, break)) ) } \\ Harry J. Smith, Sep 07 2009

Primes p = prime(k) = A000040(k) such that A359440(k+3) >= 3. - Peter Munn, Jan 09 2023

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

A064103 Primes p = p(k) such that p(k) + p(k+9) = p(k+1) + p(k+8) = p(k+2) + p(k+7) = p(k+3) + p(k+6) = p(k+4) + p(k+5).

Original entry on oeis.org

13, 139, 6091, 19843, 51787, 55793, 113143, 179029, 205157, 302551, 346361, 460949, 895799, 970447, 1150651, 1180847, 1697257, 1929553, 2334781, 2580631, 2797447, 3056561, 3086009, 3416717, 3598943, 4024667, 4026107, 4067123, 4077583, 4389503, 4541083, 4790503

Offset: 1

Robert G. Wilson v, Sep 17 2001

			13 + 47 = 17 + 43 = 19 + 41 = 23 + 37 = 29 + 31.

Cf. A000040, A022885, A064101, A359440.

a = {0, 0, 0, 0, 0, 0, 0, 0, 0, 0}; Do[ a = Delete[ a, 1 ]; a = Append[ a, Prime[ n ] ]; If[ a[ [ 1 ] ] + a[ [ 10 ] ] == a[ [ 2 ] ] + a[ [ 9 ] ] == a[ [ 3 ] ] + a[ [ 8 ] ] == a[ [ 4 ] ] + a[ [ 7 ] ] == a[ [ 5 ] ] + a[ [ 6 ] ], Print[ a[ [ 1 ] ] ] ], {n, 1, 3 10^5} ]

Primes p = prime(k) = A000040(k) such that A359440(k+4) >= 4. - Peter Munn, Jan 13 2023

More terms from Sean A. Irvine, Jun 11 2023

A064104 Primes p = p(k) such that p(k) + p(k+11) = p(k+1) + p(k+10) = p(k+2) + p(k+9) = p(k+3) + p(k+8) = p(k+4) + p(k+7) = p(k+5) + p(k+6).

Original entry on oeis.org

137, 55787, 113131, 179021, 895789, 1150649, 3086003, 4026103, 4077559, 8021753, 8750857, 12577063, 14355559, 19136527, 19412863, 20065961, 21865339, 22633141, 25880177, 30404971, 33926159, 38202173, 41905891, 42925699

Offset: 1

Robert G. Wilson v, Sep 17 2001

			137 + 193 = 139 + 191 = 149 + 181 = 151 + 179 = 157 + 173 = 163 + 167.

Cf. A000040, A022885, A064101, A359440.

a = {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}; Do[ a = Delete[ a, 1 ]; a = Append[ a, Prime[ n ] ]; If[ a[ [ 1 ] ] + a[ [ 12 ] ] == a[ [ 2 ] ] + a[ [ 11 ] ] == a[ [ 3 ] ] + a[ [ 10 ] ] == a[ [ 4 ] ] + a[ [ 9 ] ] == a[ [ 5 ] ] + a[ [ 8 ] ] == a[ [ 6 ] ] + a[ [ 7 ] ], Print[ a[ [ 1 ] ] ] ], {n, 1, 10^6} ]
okQ[n_]:=Length[Union[Take[n,6]+Reverse[Take[n,-6]]]]==1; Transpose[ Select[Partition[Prime[Range[2700000]],12,1],okQ]][[1]] (* Harvey P. Dale, Apr 25 2011 *)

Primes p = prime(k) = A000040(k) such that A359440(k+5) >= 5. - Peter Munn, Jan 13 2023

A367848 Lengths >= 2 of symmetrical subsequences within the prime gaps sequence.

Original entry on oeis.org

2, 3, 5, 5, 3, 9, 5, 2, 3, 3, 3, 5, 3, 3, 5, 2, 11, 2, 3, 3, 2, 3, 2, 3, 2, 3, 5, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 2, 5, 2, 2, 3, 7, 3, 2, 3, 3, 5, 5, 7, 3, 3, 5, 2, 2, 3, 5, 3, 3, 3, 2, 5, 2, 3, 2, 2, 3, 7, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 2, 3, 2, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 5

Offset: 1

Tamas Sandor Nagy, Dec 02 2023

nonn

Points in the primes gap sequence (A001223) are taken successively at a term and halfway between terms.
The lengths here are of subsequences made of 2 or more symmetrically placed, consecutive prime gaps around such a point.
Some points only have a subsequence of length 0 or 1 around them and they are ignored.
Will all odd numbers appear in this sequence?
Do the terms have a long-term average?

			The first lengths are as follows, around midpoints marked with ".",
  Gaps:  1   2   2   4   2   4   2    = A001223
             \_._/                  length 2 = a(1)
                 \___.___/          length 3 = a(2)
                 \_______._______/  length 5 = a(3)

Cf. A000040, A001223, A081235, A346399, A359440.

diff(v) = vector(#v-1, i, v[i+1]-v[i]);
issym(v) = if (#v>1, for (j=1, #v\2, if (v[j] != v[#v-j+1], return(0))); return(1));
lista(nn) = my(v = diff(primes(nn))); for (len=2, #v, for (i=0, len\2, my(w = vector(len-2*i, j, v[i+j])); if (issym(w), print1(#w, ", "); break););); \\ Michel Marcus, Dec 05 2023

cp's OEIS Frontend

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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A064101 Primes p = prime(k) such that prime(k) + prime(k+5) = prime(k+1) + prime(k+4) = prime(k+2) + prime(k+3).

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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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A064102 Primes p = prime(k) such that prime(k) + prime(k+7) = prime(k+1) + prime(k+6) = prime(k+2) + prime(k+5) = prime(k+3) + prime(k+4).

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

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A064103 Primes p = p(k) such that p(k) + p(k+9) = p(k+1) + p(k+8) = p(k+2) + p(k+7) = p(k+3) + p(k+6) = p(k+4) + p(k+5).

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A064104 Primes p = p(k) such that p(k) + p(k+11) = p(k+1) + p(k+10) = p(k+2) + p(k+9) = p(k+3) + p(k+8) = p(k+4) + p(k+7) = p(k+5) + p(k+6).

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