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 1-4 of 4 results.

A268926 Primes 6k + 1 preceding the maximal gaps in A268925.

Original entry on oeis.org

7, 19, 43, 241, 1327, 4363, 7789, 22189, 24247, 38461, 40237, 69499, 480169, 1164607, 1207699, 1467937, 1526659, 3975721, 11962651, 14466637, 19097257, 30097861, 39895309, 198389311, 303644227, 393202123, 485949253, 680676109, 1917214927, 3868900621, 4899889741, 6957509653, 7599382573
Offset: 1

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Author

Alexei Kourbatov, Feb 15 2016

Keywords

Comments

Subsequence of A002476 and A330854.
A268925 lists the corresponding record gap sizes. See more comments there.

Examples

			The first two primes of the form 6k+1 are 7 and 13, so a(1)=7. The next prime of this form is 19; the gap 19-13 is not a record so nothing is added to the sequence. The next prime of this form is 31 and the gap 31-19=12 is a new record, so a(2)=19.

Links

Crossrefs

Cf. A002476, A268925, A268927, A330853, A330854.

Programs

  • PARI
    re=0; s=7; forprime(p=13, 1e8, if(p%6!=1, next); g=p-s; if(g>re, re=g; print1(s", ")); s=p)

Formula

a(n) = A268927(n) - A268925(n). - Alexei Kourbatov, Jun 21 2020

A268927 Primes 6k + 1 at the end of the maximal gaps in A268925.

Original entry on oeis.org

13, 31, 61, 271, 1381, 4423, 7867, 22273, 24337, 38557, 40351, 69661, 480343, 1164799, 1207903, 1468189, 1526929, 3976003, 11962963, 14466967, 19097593, 30098239, 39895771, 198389797, 303644749, 393202651, 485949787, 680676709, 1917215533, 3868901233, 4899890383, 6957510319, 7599383353
Offset: 1

Views

Author

Alexei Kourbatov, Feb 15 2016

Keywords

Comments

Subsequence of A002476 and A330855.
A268925 lists the corresponding record gap sizes. See more comments there.

Examples

			The first two primes of the form 6k+1 are 7 and 13, so a(1)=13. The next prime of this form is 19; the gap 19-13 is not a record so nothing is added to the sequence. The next prime of this form is 31 and the gap 31-19=12 is a new record, so a(2)=31.

Links

Crossrefs

Cf. A002476, A268925, A268926, A330853, A330855.

Programs

  • PARI
    re=0; s=7; forprime(p=13, 1e8, if(p%6!=1, next); g=p-s; if(g>re, re=g; print1(p", ")); s=p)

Formula

a(n) = A268925(n) + A268926(n). - Alexei Kourbatov, Jun 21 2020

A330853 First occurrences of gaps between primes 6k+1: gap sizes.

Original entry on oeis.org

6, 12, 18, 30, 24, 54, 42, 36, 48, 60, 78, 66, 72, 84, 90, 96, 114, 102, 162, 108, 126, 120, 132, 150, 138, 144, 174, 168, 156, 192, 204, 180, 198, 252, 270, 216, 222, 186, 228, 210, 240, 282, 246, 234, 276, 264, 258, 312, 330, 318, 288, 306, 294, 336, 300, 378
Offset: 1

Views

Author

Alexei Kourbatov, Apr 27 2020

Keywords

Comments

Contains A268925 as a subsequence.
Conjecture: the sequence is a permutation of all positive multiples of 6, i.e., all positive terms of A008588.
Conjecture: a(n) = O(n). See arXiv:2002.02115 (sect.7) for discussion.

Examples

			The first primes of the form 6k+1 are 7 and 13, so a(1)=13-7=6. The next prime 6k+1 is 19, and the gap 19-13=6 already occurred, so a new term is not added to the sequence. The next prime 6k+1 is 31, and the gap 31-19=12 is occurring for the first time; therefore a(2)=12.

Links

Crossrefs

Cf. A002476, A014320, A058320, A330854 (primes 6k+1 preceding the first-occurrence gaps), A330855 (primes 6k+1 at the end of the first-occurrence gaps).

Programs

  • PARI
    isFirstOcc=vector(9999,j,1); s=7; forprime(p=13,1e8, if(p%6!=1,next); g=p-s; if(isFirstOcc[g/6], print1(g", "); isFirstOcc[g/6]=0); s=p)

Formula

a(n) = A330855(n) - A330854(n).

A084162 a(n) is the length of the gap in sequence A084161.

Original entry on oeis.org

3, 8, 12, 16, 24, 32, 48, 56, 60, 68, 72, 88, 108, 128, 148, 152, 200, 224, 240, 248, 252, 260, 272, 280, 324, 360, 420, 444, 460, 516, 520, 540, 628, 684, 696, 716, 720, 744, 800, 884, 960, 1044, 1084
Offset: 0

Views

Author

Sven Simon, May 17 2003

Keywords

Comments

First occurrence maximum gaps in sequence A002313 (real primes with corresponding complex primes).
From Alexei Kourbatov, Feb 16 2016: (Start)
Dirichlet's theorem on arithmetic progressions and GRH suggest that average gaps between primes of the form 4k + 1 below x are about phi(4)*log(x). This sequence shows that the record gap ending at p grows almost as fast as phi(4)*log^2(p). Here phi(n) is A000010, Euler's totient function; phi(4)=2.
Conjecture: a(n) < phi(4)*log^2(A268963(n)); A268963 are the end-of-gap primes.
(End)
Conjecture: a(n) < phi(4)*n^2 for all n > 2. (Note the starting offset 0.) - Alexei Kourbatov, Aug 12 2017

Examples

			a(3) = 16: There are no primes p = 1 mod 4 between 73 and 89, this gap is the largest up to 89, the gap size is 16.

Links

Crossrefs

Cf. A002313, A084160, A084161 (start of gap), A268963 (end of gap); A268799, A268925, A268928.

Programs

  • Mathematica
    Reap[Print[3]; Sow[3]; r = 0; p = 5; For[q = 7, q < 10^7, q = NextPrime[q], If[Mod[q, 4] == 3, Continue[]]; g = q - p; If[g > r, r = g; Print[g] Sow[g]]; p = q]][[2, 1]] (* Jean-François Alcover, Feb 20 2019, from PARI *)
  • PARI
    print1(3); r=0; p=5; forprime(q=7, 1e7, if(q%4==3, next); g=q-p; if(g>r, r=g; print1(", "g)); p=q)
Showing 1-4 of 4 results.

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