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

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, 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, 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

Offset

1,1

Comments

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.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

V. Shevelev, Ramanujan and Labos primes, their generalizations, and classifications of primes, J. Integer Seq. 15 (2012) Article 12.5.4.

J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, J. Integer Seq. 14 (2011) Article 11.6.2.

Prog

(Haskell)
import Data.List (group)
a182426 n = a182426_list !! (n-1)
a182426_list = concatMap f $ group $ zipWith (-) (tail ips) ips where
   f xs | head xs == 1 = reverse $ enumFromTo 2 $ length xs + 1
        | otherwise    = take (length xs) $ repeat 1
   ips = map a049084 a166251_list
-- Reinhard Zumkeller, May 18 2012

Crossrefs

Cf. A166251, A182423, A182405, A164368, A194598.

Cf. A049084.

Sequence in context: A095691 A375428 A368105 * A371733 A067029 A087179

Adjacent sequences: A182423 A182424 A182425 * A182427 A182428 A182429

Keyword

nonn

Author

Vladimir Shevelev, Apr 28 2012

Extensions

Data corrected: a(49)=2.

Status

approved