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%I #37 Feb 27 2019 14:18:07
%S 3,8,12,16,24,32,48,56,60,68,72,88,108,128,148,152,200,224,240,248,
%T 252,260,272,280,324,360,420,444,460,516,520,540,628,684,696,716,720,
%U 744,800,884,960,1044,1084
%N a(n) is the length of the gap in sequence A084161.
%C First occurrence maximum gaps in sequence A002313 (real primes with corresponding complex primes).
%C From _Alexei Kourbatov_, Feb 16 2016: (Start)
%C Dirichlet's theorem on arithmetic progressions and GRH suggest that average gaps between primes of the form 4k + 1 below x are about phi(4)*log(x). This sequence shows that the record gap ending at p grows almost as fast as phi(4)*log^2(p). Here phi(n) is A000010, Euler's totient function; phi(4)=2.
%C Conjecture: a(n) < phi(4)*log^2(A268963(n)); A268963 are the end-of-gap primes.
%C (End)
%C Conjecture: a(n) < phi(4)*n^2 for all n > 2. (Note the starting offset 0.) - _Alexei Kourbatov_, Aug 12 2017
%H Alexei Kourbatov, <a href="http://arxiv.org/abs/1610.03340">On the distribution of maximal gaps between primes in residue classes</a>, arXiv:1610.03340 [math.NT], 2016.
%H Alexei Kourbatov, <a href="https://arxiv.org/abs/1709.05508">On the nth record gap between primes in an arithmetic progression</a>, arXiv:1709.05508 [math.NT], 2017; <a href="https://doi.org/10.12988/imf.2018.712103">Int. Math. Forum, 13 (2018), 65-78</a>.
%H Alexei Kourbatov, Marek Wolf, <a href="https://arxiv.org/abs/1901.03785">Predicting maximal gaps in sets of primes</a>, arXiv:1901.03785 [math.NT], 2019.
%e a(3) = 16: There are no primes p = 1 mod 4 between 73 and 89, this gap is the largest up to 89, the gap size is 16.
%t Reap[Print[3]; Sow[3]; r = 0; p = 5; For[q = 7, q < 10^7, q = NextPrime[q], If[Mod[q, 4] == 3, Continue[]]; g = q - p; If[g > r, r = g; Print[g] Sow[g]]; p = q]][[2, 1]] (* _Jean-François Alcover_, Feb 20 2019, from PARI *)
%o (PARI) print1(3); r=0; p=5; forprime(q=7, 1e7, if(q%4==3, next); g=q-p; if(g>r, r=g; print1(", "g)); p=q)
%Y Cf. A002313, A084160, A084161 (start of gap), A268963 (end of gap); A268799, A268925, A268928.
%K nonn
%O 0,1
%A _Sven Simon_, May 17 2003
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