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 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 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 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 Cf. A007521, A269514, A269515. Sequence in context: A057584 A063526 A156331 * A024700 A108576 A052207 Adjacent sequences:  A269510 A269511 A269512 * A269514 A269515 A269516 KEYWORD nonn AUTHOR Alexei Kourbatov, Feb 28 2016 STATUS approved

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Last modified August 9 21:08 EDT 2022. Contains 356026 sequences. (Running on oeis4.)