A268984 Record (maximal) gaps between primes of the form 10k + 1.

20, 30, 70, 100, 110, 120, 180, 190, 240, 280, 330, 340, 400, 410, 450, 480, 500, 540, 570, 590, 620, 640, 670, 720, 740, 770, 990, 1110, 1240, 1260, 1380, 1480, 1650, 1780, 1800, 1890, 1900, 1910, 2070

Offset

1,1

Comments

Dirichlet's theorem on arithmetic progressions suggests that average gaps between primes of the form 10k + 1 below x are about phi(10)*log(x). This sequence shows that the record gap ending at p grows almost as fast as phi(10)*log^2(p). Here phi(n) is A000010, Euler's totient function; phi(10)=4.

Conjecture: a(n) < phi(10)*log^2(A268986(n)) almost always.

A268985 lists the primes preceding the maximal gaps.

A268986 lists the corresponding primes at the end of the maximal gaps.

Links

Table of n, a(n) for n=1..39.

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.

Example

The first two primes of the form 10k + 1 are 11 and 31, so a(1)=31-11=20. The next primes of this form are 41, 61, 71; the gaps 41-31, 61-41, 71-61 are not records so nothing is added to the sequence. The next prime of this form is 101 and the gap 101-71=30 is a new record, so a(2)=30.

Prog

(PARI) re=0; s=11; forprime(p=31, 1e8, if(p%10!=1, next); g=p-s; if(g>re, re=g; print1(g", ")); s=p)

Crossrefs

Cf. A030430, A268985, A268986.

Sequence in context: A256227 A142342 A008444 * A359002 A066214 A285494

Adjacent sequences: A268981 A268982 A268983 * A268985 A268986 A268987

Keyword

nonn

Author

Alexei Kourbatov, Feb 16 2016

Status

approved