A268928 -id:A268928 - cp's OEIS frontend

A268929 Primes 6k - 1 preceding the maximal gaps in A268928.

5, 29, 113, 197, 521, 1109, 1733, 6389, 7349, 35603, 148517, 180797, 402593, 406907, 2339039, 5521721, 11157989, 20831267, 22440701, 27681263, 73451723, 241563407, 953758109, 1444257671, 1917281213, 6822753629, 15867286361, 28265029631, 40841579819, 177858259463

Offset: 1

Alexei Kourbatov, Feb 15 2016

nonn

Subsequence of A007528 and A334544.

A268928 lists the corresponding record gap sizes. See more comments there.

			The first two primes of the form 6k-1 are 5 and 11, so a(1)=5. The next primes of this form are 17, 23, 29; the gaps 17-11 = 23-17 = 29-23 are not records so nothing is added to the sequence. The next prime of this form is 41 and the gap 41-29=12 is a new record, so a(2)=29.

Alexei Kourbatov, Table of n, a(n) for n = 1..37
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.

Cf. A007528, A268928, A268930, A334543, A334544.

re=0; s=5; forprime(p=11, 1e8, if(p%6!=5, next); g=p-s; if(g>re, re=g; print1(s", ")); s=p)

a(n) = A268930(n) - A268928(n). - Alexei Kourbatov, Jun 15 2020.

A268930 Primes 6k - 1 at the end of the maximal gaps in A268928.

Original entry on oeis.org

11, 41, 131, 227, 557, 1151, 1787, 6449, 7433, 35729, 148667, 180959, 402761, 407153, 2339297, 5522039, 11158331, 20831621, 22441073, 27681671, 73452191, 241563941, 953758661, 1444258271, 1917281867, 6822754391, 15867287129, 28265030429, 40841580683, 177858260357

Offset: 1

Alexei Kourbatov, Feb 15 2016

nonn

Subsequence of A007528 and A334545.

A268928 lists the corresponding record gap sizes. See more comments there.

			The first two primes of the form 6k-1 are 5 and 11, so a(1)=11. The next primes of this form are 17, 23, 29; the gaps 17-11 = 23-17 = 29-23 are not records so nothing is added to the sequence. The next prime of this form is 41 and the gap 41-29=12 is a new record, so a(2)=41.

Alexei Kourbatov, Table of n, a(n) for n = 1..37
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.

Cf. A007528, A268928, A268929, A334543, A334545.

re=0; s=5; forprime(p=11, 1e8, if(p%6!=5, next); g=p-s; if(g>re, re=g; print1(p", ")); s=p)

a(n) = A268928(n) + A268929(n). - Alexei Kourbatov, Jun 15 2020.

A334543 First occurrences of gaps between primes 6k - 1: gap sizes.

Original entry on oeis.org

6, 12, 18, 30, 24, 36, 42, 54, 48, 60, 84, 66, 78, 72, 126, 90, 102, 108, 114, 96, 120, 150, 138, 162, 132, 144, 168, 246, 156, 180, 186, 240, 204, 192, 216, 198, 210, 174, 258, 252, 222, 234, 228, 318, 282, 264, 276, 342, 306, 294, 312, 270, 354, 372

Offset: 1

Alexei Kourbatov, May 05 2020

nonn

Contains A268928 as a subsequence. First differs from A268928 at a(5)=24.
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.

			The first two primes of the form 6k-1 are 5 and 11, so a(1)=11-5=6. The next primes of this form are 17, 23, 29; the gaps 17-11 = 23-17 = 29-23 have size 6 which already occurred before; so nothing is added to the sequence. The next prime of this form is 41 and the gap 41-29=12 has not occurred before, so a(2)=12.

Alexei Kourbatov, Table of n, a(n) for n = 1..160
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. A007528, A014320, A058320, A268928, A330853, A334544, A334545.

isFirstOcc=vector(9999,j,1); s=5; forprime(p=11,1e8,if(p%6!=5,next); g=p-s; if(isFirstOcc[g/6], print1(g", "); isFirstOcc[g/6]=0); s=p)

a(n) = A334545(n) - A334544(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

Sven Simon, May 17 2003

nonn

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

			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.

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, Marek Wolf, Predicting maximal gaps in sets of primes, arXiv:1901.03785 [math.NT], 2019.

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

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 *)

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)

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A268929 Primes 6k - 1 preceding the maximal gaps in A268928.

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A268930 Primes 6k - 1 at the end of the maximal gaps in A268928.

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A334543 First occurrences of gaps between primes 6k - 1: gap sizes.

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

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