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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
Alexei Kourbatov, On the distribution of maximal gaps between primes in residue classes, arXiv:1610.03340 [math.NT], 2016.
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 + 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
Sequence in context: A344333 A034127 A229466 * A328259 A005052 A057344
KEYWORD
nonn
AUTHOR
Alexei Kourbatov, Feb 20 2016
STATUS
approved

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Last modified July 25 00:10 EDT 2024. Contains 374585 sequences. (Running on oeis4.)