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A269261 Record (maximal) gaps between primes of the form 10k + 9. 2
10, 30, 80, 100, 110, 120, 170, 180, 190, 240, 270, 280, 290, 330, 360, 370, 500, 510, 610, 620, 630, 670, 700, 730, 840, 870, 950, 990, 1020, 1130, 1220, 1280, 1320, 1610, 1770, 1910, 2450 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Dirichlet's theorem on arithmetic progressions suggests that average gaps between primes of the form 10k + 9 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(A269263(n)) almost always.

A269262 lists the primes preceding the maximal gaps.

A269263 lists the corresponding primes at the end of the maximal gaps.

LINKS

Table of n, a(n) for n=1..37.

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.

EXAMPLE

The first two primes of the form 10k + 9 are 19 and 29, so a(1)=29-19=10. The next prime of this form is 59 and the gap 59-29=30 is a new record, so a(2)=30.

PROG

(PARI) re=0; s=19; forprime(p=29, 1e8, if(p%10!=9, next); g=p-s; if(g>re, re=g; print1(g", ")); s=p)

CROSSREFS

Cf. A030433, A269262, A269263.

Sequence in context: A124080 A034127 A229466 * A005052 A057344 A115134

Adjacent sequences:  A269258 A269259 A269260 * A269262 A269263 A269264

KEYWORD

nonn

AUTHOR

Alexei Kourbatov, Feb 20 2016

STATUS

approved

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Last modified November 18 18:33 EST 2018. Contains 317324 sequences. (Running on oeis4.)