A235925 Primes p with q = prime(p) - p + 1 and r = prime(q) - q + 1 both prime.

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

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

Zhi-Wei 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

Zhi-Wei Sun, Jan 17 2014

Status

approved