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A084162
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a(n) is the length of the gap in sequence A084161.
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2
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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
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0,1
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COMMENTS
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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 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
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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