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 A269519 Record (maximal) gaps between primes of the form 8k + 7. 2
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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. LINKS 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. EXAMPLE 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. PROG (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) CROSSREFS Cf. A007522, A269520, A269521. Sequence in context: A349241 A120142 A110228 * A175161 A045875 A046434 Adjacent sequences: A269516 A269517 A269518 * A269520 A269521 A269522 KEYWORD nonn AUTHOR Alexei Kourbatov, Feb 28 2016 STATUS approved

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Last modified March 20 06:22 EDT 2023. Contains 361359 sequences. (Running on oeis4.)