

A269513


Record (maximal) gaps between primes of the form 8k + 5.


2



8, 16, 40, 48, 56, 64, 72, 80, 88, 96, 112, 128, 144, 192, 216, 224, 264, 288, 296, 360, 368, 440, 456, 480, 608, 616, 672, 752, 760, 856, 912, 920, 960, 1128, 1176, 1216, 1424, 1432, 1440, 1464, 1480, 1552, 1728, 1872
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OFFSET

1,1


COMMENTS

Dirichlet's theorem on arithmetic progressions suggests that average gaps between primes of the form 8k + 5 below x are about phi(8)*log(x). This sequence shows that the record gap ending at p grows almost as fast as phi(8)*log^2(p). Here phi(n) is A000010, Euler's totient function; phi(8)=4.
Conjecture: a(n) < phi(8)*log^2(A269515(n)) almost always.
A269514 lists the primes preceding the maximal gaps.
A269515 lists the corresponding primes at the end of the maximal gaps.


LINKS



EXAMPLE

The first two primes of the form 8k + 5 are 5 and 13, so a(1)=135=8. The next prime of this form is 29 and the gap 2913=16 is a new record, so a(2)=16.


PROG

(PARI) re=0; s=5; forprime(p=13, 1e8, if(p%8!=5, next); g=ps; if(g>re, re=g; print1(g", ")); s=p)


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



