

A235925


Primes p with q = prime(p)  p + 1 and r = prime(q)  q + 1 both prime.


8



2, 3, 5, 17, 23, 41, 61, 83, 181, 271, 311, 337, 757, 953, 1277, 1451, 1753, 1777, 2027, 2081, 2341, 2707, 2713, 2749, 2819, 2861, 2879, 2909, 2971, 3121, 3163, 3329, 3697, 3779, 3833, 3881, 3907, 4027, 4051, 4129, 4363, 4549, 5333, 5483, 5659, 5743, 5813, 5897, 6029, 6053
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OFFSET

1,1


COMMENTS

By the conjecture in A235924, this sequence should have infinitely many terms.
Conjecture: For any integer m > 1, there are infinitely many chains p(1) < p(2) < ... < p(m) of m primes with p(k+1) = prime(p(k))  p(k) + 1 for all 0 < k < m.


LINKS

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


EXAMPLE

a(1) = 2 since prime(2)  2 + 1 = 2 is prime.
a(2) = 3 since prime(3)  3 + 1 = 3 is prime.
a(3) = 5 since 5, prime(5)  5 + 1 = 7 and prime(7)  7 + 1 = 11 are all prime.


MATHEMATICA

f[n_]:=Prime[n]n+1
n=0; Do[If[PrimeQ[f[Prime[k]]]&&PrimeQ[f[f[Prime[k]]]], n=n+1; Print[n, " ", Prime[k]]], {k, 1, 1000}]
bpQ[n_]:=Module[{q=Prime[n]n+1}, AllTrue[{q, Prime[q]q+1}, PrimeQ]]; Select[Prime[Range[800]], bpQ](* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Nov 07 2014 *)


CROSSREFS

Cf. A000040, A234695, A235924.
Sequence in context: A240679 A180474 A155978 * A106859 A055472 A077499
Adjacent sequences: A235922 A235923 A235924 * A235926 A235927 A235928


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Jan 17 2014


STATUS

approved



