A268927 -id:A268927 - cp's OEIS frontend

A330855 Primes 6k + 1 at the end of first-occurrence gaps in A330853.

13, 31, 61, 271, 307, 1381, 1531, 1987, 2437, 4423, 7867, 10243, 16831, 22273, 24337, 38557, 40351, 43543, 69661, 75511, 100927, 119047, 171403, 195691, 204301, 250423, 480343, 577807, 590593, 1164799, 1207903, 1278997, 1382419, 1468189, 1526929, 1890019, 2314591

Offset: 1

Alexei Kourbatov, Apr 27 2020

nonn

Subsequence of A002476. Contains A268927 as a subsequence. First differs from A268927 at a(5)=307.

A330853 lists the corresponding gap sizes; see more comments there.

			The first two primes of the form 6k+1 are 7 and 13, so a(1)=13. 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)=31.

Alexei Kourbatov, Table of n, a(n) for n = 1..135
Alexei Kourbatov and Marek Wolf, On the first occurrences of gaps between primes in a residue class, arXiv preprint arXiv:2002.02115 [math.NT], 2020.

Cf. A002476, A014320, A058320, A268927, A330853 (first-occurrence gap sizes), A330854 (primes beginning the first-occurrence gaps).

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

a(n) = A330853(n) + A330854(n).

A268925 Record (maximal) gaps between primes of the form 6k + 1.

Original entry on oeis.org

6, 12, 18, 30, 54, 60, 78, 84, 90, 96, 114, 162, 174, 192, 204, 252, 270, 282, 312, 330, 336, 378, 462, 486, 522, 528, 534, 600, 606, 612, 642, 666, 780, 810, 894, 1002

Offset: 1

Alexei Kourbatov, Feb 15 2016

Dirichlet's theorem on arithmetic progressions and the GRH suggest that average gaps between primes of the form 6k + 1 below x are about phi(6)*log(x). This sequence shows that the record gap ending at p grows almost as fast as phi(6)*log^2(p). Here phi(n) is A000010, Euler's totient function; phi(6)=2.
Conjecture: a(n) < phi(6)*log^2(A268927(n)) almost always.
Conjecture: phi(6)*n^2/6 < a(n) < phi(6)*n^2 almost always. - Alexei Kourbatov, Nov 27 2019

			The first two primes of the form 6k+1 are 7 and 13, so a(1)=13-7=6. 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; the gap 31-19=12 is a new record, so a(2)=12.

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. A002476, A268926 (primes preceding the maximal gaps), A268927 (primes at the end of maximal gaps), A330853, A330854.

re = 0; s = 7; Reap[For[p = 13, p < 10^8, p = NextPrime[p], If[Mod[p, 6] != 1, 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 *)
records[n_]:=Module[{ri=n,m=0,rcs={},len},len=Length[ri];While[len>0,If[ First[ri]>m,m=First[ri];AppendTo[rcs,m]];ri=Rest[ri];len--]; rcs]; records[ Differences[Select[6*Range[0,3*10^6]+1,PrimeQ]]] (* the program generates the first 30 terms of the sequence. *) (* Harvey P. Dale, Dec 19 2021 *)

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

a(n) = A268927(n) - A268926(n). - Alexei Kourbatov, Jun 21 2020

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

Alexei Kourbatov, Feb 15 2016

nonn

Subsequence of A002476 and A330854.

A268925 lists the corresponding record gap sizes. See more comments there.

			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.

Alexei Kourbatov, Table of n, a(n) for n = 1..36
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.

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

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)

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

cp's OEIS Frontend

A330855 Primes 6k + 1 at the end of first-occurrence gaps in A330853.

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A268925 Record (maximal) gaps between primes of the form 6k + 1.

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

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