cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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.

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

Views

Author

Tomáš Brada, Aug 09 2020

Keywords

Examples

			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.

Crossrefs

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

Views

Author

Tomáš Brada, Sep 20 2020

Keywords

Links

Crossrefs

Cf. A000040, A055381, A055382, A064101, A081235, A175309, A335044, A335394, A336967, A336968, A359440.

Formula

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

A263171 Smallest prime starting a sequence of 4 consecutive odd primes such that the center of the symmetrical gaps is 2n.

Original entry on oeis.org

7, 5, 251, 353, 137, 2393, 109, 1931, 1753, 883, 3733, 7351, 12007, 2969, 8887, 27697, 1321, 22811, 38377, 62987, 183823, 15679, 124001, 180563, 45887, 48677, 100847, 178693, 152993, 557087, 34057, 367949, 294551, 134507, 173357, 1802407, 531359, 1134311, 933067
Offset: 1

Views

Author

Michel Lagneau, Oct 11 2015

Keywords

Comments

The sequence is generalizable with primes starting a sequence of 2k consecutive odd primes.
Conjecture: a(n) exists for all n>0.

Examples

			a(2)=5 because the 4 consecutive primes 5, 7, 11, 13 have gaps 2, 4, 2, which is symmetric about its center 4 = 2*2.

Crossrefs

Cf. A001223, A055380, A055381, A055382, A175309.

Programs

  • Maple
    with(numtheory):nn:=500000:l:=2:T:=array(1..2*l-1)):
for n from 1 to 35 do:ii:=0:
  for k from 1 to nn while(ii=0) do:
      lst:={}:lst1:={}:
       for m from 1 to 2*l do:
        lst:=lst union {ithprime(k+m-1)}
       od:
         for p from 1 to 2*l do:
          lst1:=lst1 union {lst[p]+lst[2*l-p+1]}
         od:
            n0:=nops(lst1):
            if n0=1
            then
           for a from 1 to 2*l-1 do:
           T[a]:=lst[a+1]-lst[a]:
           od:
           if T[2]=2*n then ii:=1:printf(`%d, `,lst[1]):
           else fi :fi:
           od :
          od:
  • PARI
    a(n) = {pa = 3; pb = 5; pc = 7; forprime(p=8, , if (((pc-pb) == 2*n) && ((pb-pa) == (p-pc)), return(pa)); pa = pb; pb = pc; pc = p;);} \\ Michel Marcus, Oct 16 2015

A267028 P(n,k) is an array read by rows, with n > 0 and k=1..5, where row n gives the chain of 5 consecutive primes {p(i), p(i+1), p(i+2), p(i+3), p(i+4)} having the symmetrical property p(i) + p(i+4) = p(i+1) + p(i+3) = 2*p(i+2) for some index i.

Original entry on oeis.org

18713, 18719, 18731, 18743, 18749, 25603, 25609, 25621, 25633, 25639, 28051, 28057, 28069, 28081, 28087, 30029, 30047, 30059, 30071, 30089, 31033, 31039, 31051, 31063, 31069, 44711, 44729, 44741, 44753, 44771, 76883, 76907, 76913, 76919, 76943
Offset: 1

Views

Author

Michel Lagneau, Feb 23 2016

Keywords

Comments

a(3 + 5*(n-1)) = A051795(n).
The immediate objective of the sequence is to examine symmetrical properties in the array P(n,k). It is interesting to note that the results with the dimension 5 are generalizable to the dimensions 7, 9, ...
Notation:
We introduce the following function S(i,j) where row i is defined by {P(i,k)} and row j is defined by {P(j,k)}, k = 1..5. Let S(i, j) = 1 if P(i,1) + P(j,5) = P(i,2) + P(j,4) = P(i,3) + P(j,3), otherwise 0.
Conjecture:
For each integer n, there exists an infinite sequence of integers b(n,m), m = 1, 2, ... such that S(n, b(n,m)) = 1.
The following table gives the first values b(n,m).
Notation in the table: "PS" = primitive sequence.
+----+------------------------------------------------+-----------+
| n | sequences b(n,m), m=1,2,... of index |included in|
+----+------------------------------------------------+-----------+
| 1 | 1, 2, 3, 5, 8, 9, 10, 12, 15, 16, 17, 18, ... | PS |
| 2 | 2, 3, 5, 8, 9, 10, 12, 15, 16, 17, 18, 19, ...| {b(1,m)} |
| 3 | 3, 5, 8, 9, 10, 12, 15, 16, 17, 18, 19, ... | {b(1,m)} |
| 4 | 4, 6, 11, 13, 14, 21, 28, 35, 39, 57, 59, ... | PS |
| 5 | 5, 8, 9, 10, 12, 15, 16, 17, 18, 19, 22, ... | {b(1,m)} |
| 6 | 6, 11, 13, 14, 21, 35, 39, 57, 59, 63, 67, ...| {b(4,m)} |
| 7 | 7, 30, 52, 55, 73, 74, 115, 159, 177, 183, ...| PS |
| 8 | 8, 9, 10, 12, 15, 16, 17, 18, 19, 22, 23, ... | {b(1,m)} |
| 9 | 9, 10, 12, 15, 16, 17, 18, 19, 22, 23, 24, ...| {b(1,m)} |
| 10 | 10, 12, 15, 16, 17, 18, 19, 22, 23, 24, 26, ...| {b(1,m)} |
| 11 | 11, 13, 14, 21, 28, 35, 39, 57, 59, 63, 67, ...| {b(4,m)} |
| 12 | 12, 15, 16, 17, 18, 19, 22, 23, 24, 26, 27, ...| {b(1,m)} |
| 13 | 13, 14, 21, 28, 35, 39, 57, 59, 63, 67, 70, ...| {b(4,m)} |
| .. | ... | ... |
| 20 | 20, 43, 56, 96, 113, 131, 135, 156, 196, ... | PS |
| 25 | 21, 33, 37, 38, 40, 47, 48, 65, 76, 79, 83, ...| PS |
...
Example: S(7, 30) = 1.
We observe primitive sequences {b(n,m)} for n = {1, 4, 7, 20, 25, ...}.
(A primitive sequence is a sequence which is not included in another.)
Properties:
(1) S(i, i)= 1 for all i;
(2) S(i, j) = 1 => S(j, i) = 1;
(3) S(i, j) = 1 and S(j, L) = 1 => S(i, L) = 1.
Example:
For n = 1, {P(1,k)} = {18713, 18719, 18731, 18743, 18749};
we choose, for instance, b(1,2) = 3 => for n = 3, {C(3,k)} = {28051, 28057, 28069, 28081, 28087};
S(1,3) = 1 because 18713 + 28087 = 18719 + 28081 = 18731 + 28069 = 18743 + 28057 = 18749 + 28051 = 46800.
In order to find the index L for satisfying the property (3), we choose, for instance, the index b(3,2) = 8 => for n = 8, {P(8,k)} = {97423, 97429, 97441, 97453, 97459} and S(3, 8) = 1 because 28051 + 97459 = 28057 + 97453 = 28069 + 97441 = 28081 + 97429 = 28087 + 97423 = 125510.
Conclusion: S(1, 3) = 1 and S(3, 8) = 1 => S(1, 8) = 1 with 18713 + 97459 = 18719 + 97453 = 18731 + 97441 = 18743 + 97429 = 18749 + 97423 = 116172.

Examples

			The first row is [18713, 18719, 18731, 18743, 18749] because 18713 + 18749 = 18719 + 18743 = 2*18731 = 37462.
The array starts with:
  [18713, 18719, 18731, 18743, 18749]
  [25603, 25609, 25621, 25633, 25639]
  [28051, 28057, 28069, 28081, 28087]
  ...

Crossrefs

Cf. A051795, A055380, A055382.

Programs

  • Maple
    U:=array(1..50,1..5):W:=array(1..2):kk:=0:
for n from 4 to 10000 do:
   for m from 2 by -1 to 1 do:
      q:=ithprime(n-m)+ithprime(n+m):W[m]:=q:
    od:
    if W[1]=W[2] and W[1]=2*ithprime(n) then
    kk:=kk+1:U[kk,1]:=ithprime(n-2):
    U[kk,2]:=ithprime(n-1):U[kk,3]:=ithprime(n):
    U[kk,4]:=ithprime(n+1):U[kk,5]:=ithprime(n+2):
    else fi:od:print(U):
    for i from 1 to kk do:
     for j from i+1 to kk do:
      s1:=U[i,1]+U[j,5]:
      s2:=U[i,2]+U[j,4]:
      s3:=U[i,3]+U[j,3]:
      s4:=U[i,4]+U[j,2]:
      s5:=U[i,5]+U[j,1]:
     if s1=s2 and s2=s3 and s3=s4 and s4=s5
     then
     printf("%d %d \n",i,j):
     else fi:
     od:
  od:
Previous Showing 11-14 of 14 results.

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