A268800 -id:A268800 - cp's OEIS frontend

A268799 Record (maximal) gaps between primes of the form 4k + 3.

4, 8, 12, 20, 24, 36, 40, 56, 60, 64, 68, 112, 120, 132, 144, 156, 168, 176, 184, 200, 240, 256, 272, 280, 296, 356, 396, 444, 452, 480, 532, 616, 620, 672, 692, 708, 840, 864, 896, 916, 1004

Offset: 1

Alexei Kourbatov, Feb 13 2016

nonn

Dirichlet's theorem on arithmetic progressions and GRH suggest that average gaps between primes of the form 4k + 3 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(A268801(n)) almost always.
Conjecture: a(n) < phi(4)*n^2 for all n>2. - Alexei Kourbatov, Aug 12 2017

			The first two primes of the form 4k+3 are 3 and 7, so a(1)=7-3=4. 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)=8.

Alexei Kourbatov, On the distribution of maximal gaps between primes in residue classes, arXiv:1610.03340 [math.NT], 2016.
Alexei Kourbatov, On the nth record gap between primes in an arithmetic progression, arXiv:1709.05508 [math.NT], 2017; Int. Math. Forum, 13 (2018), 65-78.
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.

Cf. A002145, A084161.

Corresponding primes: A268800 (lower ends), A268801 (upper ends).

re = 0; s = 3; Reap[For[p = 7, p < 10^8, p = NextPrime[p], If[Mod[p, 4] != 3, Continue[]]; g = p - s; If[g > re, re = g; Print[g]; Sow[g]]; s = p]][[2, 1]] (* Jean-François Alcover, Dec 12 2018, from PARI *)

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

Alexei Kourbatov, Feb 13 2016

nonn

Subsequence of A002145.

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

			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.

Alexei Kourbatov, Table of n, a(n) for n = 1..41
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.

Cf. A002145, A084161, A268799, A268800.

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)

cp's OEIS Frontend

A268799 Record (maximal) gaps between primes of the form 4k + 3.

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