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.

Showing 1-6 of 6 results.

A329158 Let P1>=3, P2, P3 be consecutive primes, with P3-P2=2. a(n)=(P2+P3)/12 when P2-P1 sets a record.

Original entry on oeis.org

1, 2, 5, 25, 87, 192, 500, 1158, 1668, 4217, 4713, 5955, 17127, 28905, 61838, 76967, 96147, 139725, 260342, 1061923, 1205080, 4663498, 8871842, 11732765, 32534740, 42313103, 77638122, 92523718, 282054523, 728833340, 2940948542, 3344803093, 11810906035
Offset: 1

Views

Author

Hugo Pfoertner, Nov 06 2019

Keywords

Comments

6*a(n)-1, 6*a(n)+1 are twin primes such that the prime gap immediately preceding 6*a(n)-1 sets a record. The corresponding gap lengths are provided in A329159.

Crossrefs

Cf. A002822, A084105, A329159, A329160, A329161, A329164, A329165.

Programs

  • PARI
    p1=3;p2=5;r=0;forprime(p3=7,1e9,if(p3-p2==2,d=p2-p1;if(d>r,r=d;print1((p2+p3)/12,", ")));p1=p2;p2=p3)

A329159 2*a(n) are the lengths of record prime gaps immediately preceding twin primes.

Original entry on oeis.org

1, 2, 3, 5, 6, 11, 14, 15, 17, 20, 24, 26, 30, 35, 39, 42, 45, 50, 66, 68, 74, 75, 89, 98, 104, 107, 117, 123, 144, 159, 165, 180, 183, 192, 195
Offset: 1

Views

Author

Hugo Pfoertner, Nov 06 2019

Keywords

Comments

The corresponding locations are provided in A329158. The gap 2 -> 3 before the pair 3, 5 is excluded.

Crossrefs

Cf. A329158, A329160, A329161, A329164, A329165.

Programs

  • PARI
    p1=3;p2=5;r=0;forprime(p3=7,1e9,if(p3-p2==2,d=p2-p1;if(d>r,r=d;print1(d/2,", ")));p1=p2;p2=p3)

Extensions

a(31)-a(34) from Jinyuan Wang, Mar 01 2020
a(35) from David Trimas, Jul 26 2023

A329160 Let P1>=5, P2, P3 be consecutive primes, with P2-P1=2. a(n)=(P1+P2)/12 when P3-P2 sets a record.

Original entry on oeis.org

1, 5, 23, 33, 87, 278, 495, 1293, 2027, 2690, 4245, 6773, 13283, 24938, 28893, 44270, 67475, 139708, 224922, 315893, 971000, 1723960, 3319792, 6228255, 7013717, 13194622, 25321985, 31864375, 32163975, 65155398, 86090027, 381175405, 452803425
Offset: 1

Views

Author

Hugo Pfoertner, Nov 08 2019

Keywords

Comments

6*a(n)-1, 6*a(n)+1 are twin primes such that the prime gap immediately following 6*a(n)+1 sets a record. The corresponding gap lengths are provided in A329161.

Examples

			See A329161.

Crossrefs

Cf. A002822, A084105, A329158, A329159, A329161, A329164, A329165.

Programs

  • PARI
    p1=5; p2=7; r=0; forprime(p3=11, 1e9, if(p2-p1==2, d=p3-p2; if(d>r, r=d; print1((p1+p2)/12, ", "))); p1=p2; p2=p3)

A329161 2*a(n) are the lengths of record prime gaps immediately following twin primes.

Original entry on oeis.org

2, 3, 5, 6, 9, 12, 14, 15, 17, 21, 26, 27, 29, 30, 35, 41, 45, 50, 59, 60, 63, 71, 74, 75, 84, 92, 96, 98, 104, 110, 125, 129, 144, 155, 156, 165, 180, 201, 204, 207
Offset: 1

Views

Author

Hugo Pfoertner, Nov 08 2019

Keywords

Comments

The corresponding locations are provided in A329160, starting at the initial pair 5, 7.

Examples

			a(1) = 2 because the prime gap after 5, 7 has length 11 - 7 = 2*a(1) = 4,
a(2) = 3 because the prime gap after 29,31 has length 37 - 31 = 2*a(2) = 6, with location given by 6*A329160(2) = 30 +- 1. The gaps before all have length 4 (11,13 -> 17), (17,19 -> 23).

Crossrefs

Cf. A329158, A329159, A329160, A329164, A329165.

Programs

  • PARI
    p1=5; p2=7; r=0; forprime(p3=11, 1e9, if(p2-p1==2, d=p3-p2; if(d>r, r=d; print1(d/2, ", "))); p1=p2; p2=p3)

Extensions

a(39)-a(40) from Jinyuan Wang, Mar 01 2020

A329165 Let P1, P2, P3, P4 be consecutive primes with P2-P1=P4-P3=2. a(n)=(P3-P1)/6 when the length of the gap with no primes between the two pairs of twin primes sets a record.

Original entry on oeis.org

1, 2, 3, 5, 6, 9, 12, 17, 18, 21, 22, 23, 25, 31, 33, 35, 40, 41, 42, 47, 48, 49, 51, 52, 53, 57, 58, 62, 63, 66, 71, 75, 77, 78, 81, 83, 85, 90, 91, 93, 98, 100, 105, 108, 111, 115, 119, 123, 125, 135, 138, 148, 150, 152, 165, 170, 173, 180
Offset: 1

Views

Author

Hugo Pfoertner, Nov 07 2019

Keywords

Comments

The position of the occurrence of the n-th record is given by A329164(n)=(P1+P2)/12.

Examples

			See A329164.

Crossrefs

Cf. A053778, A329158, A329159, A329160, A329161, A329164.

Programs

  • Mathematica
    With[{s = Partition[Prime@ Range[10^5], 4, 1]}, Union@ FoldList[Max, Map[(#3 - #1)/6 & @@ # &, Select[s, #2 - #1 == #4 - #3 == 2 & @@ # &]]]] (* Michael De Vlieger, May 26 2020 *)
  • PARI
    p1=3;p2=5;p3=7;r=0;forprime(p4=11,1e9,if(p2-p1==2&&p4-p3==2,d=p3-p1;if(d>r,r=d;print1(d/6,", ")));p1=p2;p2=p3;p3=p4)

Extensions

a(27)-a(28) from Jinyuan Wang, Mar 01 2020
a(29)-a(58) found by Tomáš Brada, Natalia Makarova, May 12 2020

A329250 Let P1, P2, P3, P4 be consecutive primes, with P2 - P1 = P4 - P3 = 2. a(n) = (P1 + P2)/12 for the first occurrence of (P3 - P1)/6 = n.

Original entry on oeis.org

1, 23, 322, 1573, 495, 3407, 10498, 85067, 8113, 112912, 166302, 28893, 189052, 510548, 598532, 812752, 139708, 716182, 2582073, 4576458, 2497092, 5130198, 5761777, 25381573, 7315173, 20200532, 40629683, 33185292, 69948743, 38771927, 13194622
Offset: 1

Views

Author

Hugo Pfoertner, Nov 09 2019

Keywords

Comments

Position of first occurrence of a gap of length P3 - P2 = 6*n - 2 containing no primes, bounded by twin primes (P1,P2) below and (P3,P4) above.

Examples

			a(4) = 1573, because the 4 primes P1 = 6*1573 - 1 = 9437, P2 = 6*1573 + 1 = 9439, P3 = P1 + 6*4 = 9461, P4 = 9463 produce the first occurrence of the gap P3 - P2 = 9461 - 9439 = 6*4 - 2 = 22. See also example in A329164.

Links

Crossrefs

Cf. A329164, A329165, A329251, A329252.

Programs

  • PARI
    my(v=vector(31),p1=3,p2=5,p3=7,r=0,d);forprime(p4=11,5e8,if(p2-p1==2&&p4-p3==2,d=(p3-p1)/6;if(v[d]==0,v[d]=(p1+p2)/12));p1=p2;p2=p3;p3=p4);v
Showing 1-6 of 6 results.

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