A236119 Primes p with prime(p) - p - 1 and prime(p) - p + 1 both prime.

5, 17, 23, 41, 71, 83, 173, 293, 337, 353, 563, 571, 719, 811, 911, 953, 1201, 1483, 1579, 1877, 2081, 2089, 2309, 2579, 2749, 2803, 3329, 3343, 3511, 3691, 3779, 3851, 3881, 3907, 4021, 4049, 4093, 4657, 4813, 5051, 5179, 5333, 5519, 5591, 6053, 6547, 6841, 7151, 7723, 8209

Offset: 1

Author

Zhi-Wei Sun, Jan 19 2014

Keywords

• nonn

Comments

By the conjecture in A236097, this sequence should have infinitely many terms.

Examples

			a(1) = 5 since neither prime(2) - 2 - 1 = 0 nor prime(3) - 3 - 1 = 1 is prime, but prime(5) - 5 - 1 = 5 and prime(5) - 5 + 1 = 7 are both 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, A001359, A006512, A014574, A234695, A235925, A236097.

Programs

Mathematica

p[n_]:=PrimeQ[Prime[n]-n-1]&&PrimeQ[Prime[n]-n+1]
n=0;Do[If[p[Prime[k]],n=n+1;Print[n," ",Prime[k]]],{k,1,1100}]

PARI

s=[]; forprime(p=2, 10000, if(isprime(prime(p)-p-1) && isprime(prime(p)-p+1), s=concat(s, p))); s \\ Colin Barker, Jan 19 2014