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A008996 Increasing length runs of consecutive composite numbers (records).

%I A008996 #49 Jul 13 2025 11:07:06
%S A008996 1,3,5,7,13,17,19,21,33,35,43,51,71,85,95,111,113,117,131,147,153,179,
%T A008996 209,219,221,233,247,249,281,287,291,319,335,353,381,383,393,455,463,
%U A008996 467,473,485,489,499,513,515,531,533,539,581,587,601,651,673,715,765
%N A008996 Increasing length runs of consecutive composite numbers (records).
%C A008996 Conjecture: a(n) = O(n^2); specifically, a(n) <= n^2. - _Alexei Kourbatov_, Jan 23 2019
%H A008996 Bert Sierra, <a href="/A008996/b008996.txt">Table of n, a(n) for n = 1..82</a> (derived from A005250; first 74 terms from Jens Kruse Andersen)
%H A008996 Jens Kruse Andersen, <a href="http://primerecords.dk/primegaps/maximal.htm">Maximal Prime Gaps</a>
%H A008996 Alexei Kourbatov, <a href="https://arxiv.org/abs/1709.05508">On the nth record gap between primes in an arithmetic progression</a>, arXiv:1709.05508 [math.NT], 2017; <a href="https://doi.org/10.12988/imf.2018.712103">Int. Math. Forum, 13 (2018), 65-78</a>.
%H A008996 Alexei Kourbatov and Marek Wolf, <a href="https://arxiv.org/abs/1901.03785">Predicting maximal gaps in sets of primes</a>, arXiv preprint arXiv:1901.03785 [math.NT], 2019.
%H A008996 Thomas R. Nicely, <a href="https://faculty.lynchburg.edu/~nicely/gaps/gaplist.html">First occurrence prime gaps</a> [For local copy see A000101]
%H A008996 Thomas R. Nicely, <a href="http://dx.doi.org/10.1090/S0025-5718-99-01065-0">New maximal prime gaps and first occurrences</a>, Math. Comput. 68,227 (1999) 1311-1315.
%H A008996 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeGaps.html">Prime Gaps</a>
%H A008996 <a href="/index/Pri#gaps">Index entries for primes, gaps between</a>
%F A008996 a(n) = A005250(n+1) - 1.
%t A008996 maxGap = 1; Reap[ Do[ gap = Prime[n+1] - Prime[n]; If[gap > maxGap, Print[gap-1]; Sow[gap-1]; maxGap = gap], {n, 2, 10^8}]][[2, 1]] (* _Jean-François Alcover_, Jun 12 2013 *)
%t A008996 Module[{nn=10^8,cmps},cmps=Table[If[CompositeQ[n],1,{}],{n,nn}];DeleteDuplicates[ Rest[ Length/@ Split[cmps]],GreaterEqual]] (* The program generates the first 24 terms of the sequnece. To generate more, increase the nn constant. *) (* _Harvey P. Dale_, Sep 04 2022 *)
%o A008996 (Haskell)
%o A008996 a008996 n = a008996_list !! (n-1)
%o A008996 a008996_list = 1 : f 0 (filter (> 1) $
%o A008996                         map length $ group $ drop 3 a010051_list)
%o A008996    where f m (u : us) = if u <= m then f m us else u : f u us
%o A008996 -- _Reinhard Zumkeller_, Nov 27 2012
%Y A008996 Cf. A005250, A008950, A008995, A058320.
%Y A008996 Cf. A010051, A046933.
%K A008996 nonn,nice
%O A008996 1,2
%A A008996 Mark Cramer (m.cramer(AT)qut.edu.au), Mar 15 1996
%E A008996 More terms from _Warren D. Smith_, Dec 11 2000
%E A008996 a(40) corrected by _Bert Sierra_, Jul 12 2025