

A084162


a(n) is the length of the gap in sequence A084161.


2



3, 8, 12, 16, 24, 32, 48, 56, 60, 68, 72, 88, 108, 128, 148, 152, 200, 224, 240, 248, 252, 260, 272, 280, 324, 360, 420, 444, 460, 516, 520, 540, 628, 684, 696, 716, 720, 744, 800, 884, 960, 1044, 1084
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OFFSET

0,1


COMMENTS

First occurrence maximum gaps in sequence A002313 (real primes with corresponding complex primes).
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.
Conjecture: a(n) < phi(4)*log^2(A268963(n)); A268963 are the endofgap primes.
(End)
Conjecture: a(n) < phi(4)*n^2 for all n > 2. (Note the starting offset 0.)  Alexei Kourbatov, Aug 12 2017


LINKS



EXAMPLE

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.


MATHEMATICA

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]] (* JeanFrançois Alcover, Feb 20 2019, from PARI *)


PROG

(PARI) print1(3); r=0; p=5; forprime(q=7, 1e7, if(q%4==3, next); g=qp; if(g>r, r=g; print1(", "g)); p=q)


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



