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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

First occurrence maximum gaps in sequence A002313 (real primes with corresponding complex primes).

From Alexei Kourbatov, Feb 16 2016: (Start)

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 end-of-gap primes.

(End)

Conjecture: a(n) < phi(4)*n^2 for all n > 2. (Note the starting offset 0.) - Alexei Kourbatov, Aug 12 2017

LINKS

Table of n, a(n) for n=0..42.

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, Marek Wolf, Predicting maximal gaps in sets of primes, arXiv:1901.03785 [math.NT], 2019.

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]] (* Jean-Fran├žois Alcover, Feb 20 2019, from PARI *)

PROG

(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)

CROSSREFS

Cf. A002313, A084160, A084161 (start of gap), A268963 (end of gap); A268799, A268925, A268928.

Sequence in context: A296368 A183991 A022806 * A140103 A190374 A189758

Adjacent sequences:  A084159 A084160 A084161 * A084163 A084164 A084165

KEYWORD

nonn

AUTHOR

Sven Simon, May 17 2003

STATUS

approved

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Last modified September 20 19:15 EDT 2020. Contains 337265 sequences. (Running on oeis4.)