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A269519
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Record (maximal) gaps between primes of the form 8k + 7.
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2
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16, 24, 40, 48, 96, 184, 200, 216, 288, 296, 312, 344, 384, 456, 504, 560, 624, 744, 760, 776, 800, 824, 840, 864, 880, 896, 952, 984, 1008, 1056, 1152, 1208, 1312, 1384, 1448, 1464, 1472, 1720, 1872
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OFFSET
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1,1
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COMMENTS
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Dirichlet's theorem on arithmetic progressions suggests that average gaps between primes of the form 8k + 7 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(A269521(n)) almost always.
A269520 lists the primes preceding the maximal gaps.
A269521 lists the corresponding primes at the end of the maximal gaps.
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LINKS
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Table of n, a(n) for n=1..39.
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.
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EXAMPLE
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The first two primes of the form 8k + 7 are 7 and 23, so a(1)=23-7=16. The next primes of this form are 31, 47; the gaps 31-23 and 47-31 are not records so nothing is added to the sequence. The next prime of this form is 71 and the gap 71-47=24 is a new record, so a(2)=24.
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PROG
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(PARI) re=0; s=7; forprime(p=23, 1e8, if(p%8!=7, next); g=p-s; if(g>re, re=g; print1(g", ")); s=p)
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CROSSREFS
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Cf. A007522, A269520, A269521.
Sequence in context: A349241 A120142 A110228 * A175161 A045875 A046434
Adjacent sequences: A269516 A269517 A269518 * A269520 A269521 A269522
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KEYWORD
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nonn
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AUTHOR
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Alexei Kourbatov, Feb 28 2016
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STATUS
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approved
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