

A238814


Primes p with prime(p)  p + 1 and prime(q)  q + 1 both prime, where q is the first prime after p.


3



2, 3, 5, 13, 41, 83, 199, 211, 271, 277, 293, 307, 349, 661, 709, 743, 751, 823, 907, 1117, 1447, 1451, 1741, 1747, 2203, 2371, 2803, 2819, 2861, 2971, 3011, 3251, 3299, 3329, 3331, 3691, 3877, 4021, 4027, 4049, 4051, 4093, 4129, 4157, 4447, 4513, 4549, 4561, 4751, 4801, 5179, 5479, 5519, 5657, 5813, 6007, 6011, 6571, 7057, 7129
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OFFSET

1,1


COMMENTS

Conjecture: The sequence is infinite, in other words, A234695 contains infinitely many consecutive prime pairs prime(k) and prime(k+1).
This is motivated by the comments in A238766 and A238776, and the sequence is a subsequence of A234695.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000
ZhiWei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014.


EXAMPLE

a(1) = 2 since prime(2)  2 + 1 = 3  1 = 2 and prime(3)  3 + 1 = 5  2 = 3 are both prime.
a(2) = 3 since prime(3)  3 + 1 = 5  2 = 3 and prime(5)  5 + 1 = 11  4 = 7 are both prime.


MATHEMATICA

p[k_]:=PrimeQ[Prime[Prime[k]]Prime[k]+1]
n=0
Do[If[p[k]&&p[k+1], n=n+1; Print[n, " ", Prime[k]]], {k, 1, 914}]
Select[Prime[Range[1000]], AllTrue[{Prime[#]#+1, Prime[NextPrime[#]]NextPrime[ #]+1}, PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Aug 24 2019 *)


PROG

(PARI) step(p, k)=k++; while(k, p=nextprime(p+1)); p
p=0; forprime(r=2, 1e6, if(isprime(p++) && isprime(rp+1), q=nextprime(p+1); if(isprime(step(r, qp)q+1), print1(p", ")))) \\ Charles R Greathouse IV, Mar 06 2014


CROSSREFS

Cf. A000040, A234694, A234695, A238766, A238776.
Sequence in context: A087362 A038560 A240838 * A000756 A192241 A093999
Adjacent sequences: A238811 A238812 A238813 * A238815 A238816 A238817


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Mar 05 2014


STATUS

approved



