A182405 -id:A182405 - cp's OEIS frontend

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

Vladimir Shevelev, Apr 28 2012

nonn

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.

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.

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

Cf. A049084.

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

Data corrected: a(49)=2.

A182423 Number of primes in interval (A194598(n), A164368(n)).

Original entry on oeis.org

0, 2, 0, 1, 1, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 3, 0, 1, 0, 3, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 1, 2, 0, 0, 0, 1, 0, 2, 1, 2, 0, 0, 0, 1, 0, 0

Offset: 1

Vladimir Shevelev, Apr 28 2012

nonn

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.

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

Cf. A164368, A194598, A182405, A166251, A182426.

A217671 a(n) is the least prime of the set of the smallest n consecutive primes a(n)=q_1(n), q_2(n),..., such that between (1/2)*q_i and (1/2)q_(i+1), i=1,...,n-1, there exists a prime, or a(n)=0 if no such set of primes exists.

Original entry on oeis.org

3, 3, 3, 73, 523, 6581, 10753, 43103, 43103, 43103, 55457, 55457, 28751773, 278689963, 278689963, 784284211, 4440915607, 8340839629, 30651695947, 50246427391, 50246427391

Offset: 2

Vladimir Shevelev, Oct 10 2012

nonn

If a(N) = 0, then a(n) = 0 for n > N. Conjecture 39 in the Shevelev link says that a(n) > 0.

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

Cf. A166251, A182405, A182523, A182426.

a(14) from John W. Layman and Hans Havermann
a(15)-a(17) from Carlos Rivera and Hans Havermann
a(18)-a(20) from Hans Havermann
a(21)-a(22) from Donovan Johnson, Oct 17 2012

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

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A182423 Number of primes in interval (A194598(n), A164368(n)).

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A217671 a(n) is the least prime of the set of the smallest n consecutive primes a(n)=q_1(n), q_2(n),..., such that between (1/2)*q_i and (1/2)q_(i+1), i=1,...,n-1, there exists a prime, or a(n)=0 if no such set of primes exists.

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