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 1-3 of 3 results.

A268800 Primes 4k + 3 preceding the maximal gaps in A268799.

Original entry on oeis.org

3, 11, 31, 83, 283, 383, 1327, 2591, 7351, 7759, 11171, 11587, 31391, 46919, 147919, 288023, 360611, 425603, 507163, 666203, 1414703, 2198887, 3358151, 9287659, 11512547, 11648531, 24315047, 42453823, 145554779, 161720147, 184007671, 766668811
Offset: 1

Views

Author

Alexei Kourbatov, Feb 13 2016

Keywords

Comments

Subsequence of A002145.
A268799 lists the corresponding record gap sizes. See more comments there.

Examples

			The first two primes of the form 4k+3 are 3 and 7, so a(1)=3. The next prime of this form is 11; the gap 11-7 is not a record so no term is added to the sequence. The next prime of this form is 19; the gap 19-11=8 is a new record so a(2)=11.

Links

Crossrefs

Cf. A002145, A084161, A268799, A268801.

Programs

  • PARI
    re=0; s=3; forprime(p=7, 1e8, if(p%4!=3, next); g=p-s; if(g>re, re=g; print1(s", ")); s=p)

A268801 Primes 4k + 3 at the end of the maximal gaps in A268799.

Original entry on oeis.org

7, 19, 43, 103, 307, 419, 1367, 2647, 7411, 7823, 11239, 11699, 31511, 47051, 148063, 288179, 360779, 425779, 507347, 666403, 1414943, 2199143, 3358423, 9287939, 11512843, 11648887, 24315443, 42454267, 145555231, 161720627, 184008203, 766669427
Offset: 1

Views

Author

Alexei Kourbatov, Feb 13 2016

Keywords

Comments

Subsequence of A002145.
A268799 lists the corresponding record gap sizes. See more comments there.

Examples

			The first two primes of the form 4k+3 are 3 and 7, so a(1)=7. The next prime of this form is 11; the gap 11-7 is not a record so no term is added to the sequence. The next prime of this form is 19; the gap 19-11=8 is a new record so a(2)=19.

Links

Crossrefs

Cf. A002145, A084161, A268799, A268800.

Programs

  • PARI
    re=0; s=3; forprime(p=7, 1e8, if(p%4!=3, next); g=p-s; if(g>re, re=g; print1(p", ")); s=p)

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 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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