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

Author

Zhi-Wei Sun, Jan 17 2014

Keywords

• nonn

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.

Examples

			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.
		

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

Crossrefs

Cf. A000040, A234695, A235924.

Programs

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 *)