cp's OEIS Frontend

A182426 Lengths of runs of consecutive isolated primes beginning with A166251(n).

%I A182426 #27 Jul 08 2018 01:41:04
%S A182426 2,1,1,1,3,2,1,1,2,1,1,1,3,2,1,1,3,2,1,2,1,2,1,4,3,2,1,1,1,2,1,1,2,1,
%T A182426 1,2,1,1,2,1,1,1,1,2,1,1,1,1,2,1,2,1,2,1,1,1,1,1,1,1,2,1,4,3,2,1,1,2,
%U A182426 1,1,1,1,2,1,2,1,2,1,1,1,1,3,2,1,2,1,1,2,1,2,1,3,2,1
%N A182426 Lengths of runs of consecutive isolated primes beginning with A166251(n).
%C A182426 Theorem. If the sequence is unbounded, then there exist arbitrarily long sequences of consecutive primes p_k, p_(k+1),...,p_m such that every interval (p_i/2, p_(i+1)/2), i=k,k+1,...,m-1, contains a prime.
%H A182426 Reinhard Zumkeller, <a href="/A182426/b182426.txt">Table of n, a(n) for n = 1..10000</a>
%H A182426 V. Shevelev, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL15/Shevelev/shevelev19.html">Ramanujan and Labos primes, their generalizations, and classifications of primes</a>, J. Integer Seq. 15 (2012) Article 12.5.4.
%H A182426 J. Sondow, J. W. Nicholson, and T. D. Noe, <a href="http://arxiv.org/abs/1105.2249"> Ramanujan Primes: Bounds, Runs, Twins, and Gaps</a>, J. Integer Seq. 14 (2011) Article 11.6.2.
%o A182426 (Haskell)
%o A182426 import Data.List (group)
%o A182426 a182426 n = a182426_list !! (n-1)
%o A182426 a182426_list = concatMap f $ group $ zipWith (-) (tail ips) ips where
%o A182426    f xs | head xs == 1 = reverse $ enumFromTo 2 $ length xs + 1
%o A182426         | otherwise    = take (length xs) $ repeat 1
%o A182426    ips = map a049084 a166251_list
%o A182426 -- _Reinhard Zumkeller_, May 18 2012
%Y A182426 Cf. A166251, A182423, A182405, A164368, A194598.
%Y A182426 Cf. A049084.
%K A182426 nonn
%O A182426 1,1
%A A182426 _Vladimir Shevelev_, Apr 28 2012
%E A182426 Data corrected: a(49)=2.