A238814 Primes p with prime(p) - p + 1 and prime(q) - q + 1 both prime, where q is the first prime after p.

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

Offset: 1

Author

Zhi-Wei Sun, Mar 05 2014

Keywords

• nonn

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.

Examples

			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.
		

Links

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

Crossrefs

Cf. A000040, A234694, A234695, A238766, A238776.

Programs

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 *)

PARI

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