|
|
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
(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 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)=13-5=8. The next prime of this form is 29 and the gap 29-13=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=p-s; if(g>re, re=g; print1(g", ")); s=p)
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|