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

%I A238814 #15 Aug 24 2019 17:18:13
%S A238814 2,3,5,13,41,83,199,211,271,277,293,307,349,661,709,743,751,823,907,
%T A238814 1117,1447,1451,1741,1747,2203,2371,2803,2819,2861,2971,3011,3251,
%U A238814 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
%N A238814 Primes p with prime(p) - p + 1 and prime(q) - q + 1 both prime, where q is the first prime after p.
%C A238814 Conjecture: The sequence is infinite, in other words, A234695 contains infinitely many consecutive prime pairs prime(k) and prime(k+1).
%C A238814 This is motivated by the comments in A238766 and A238776, and the sequence is a subsequence of A234695.
%H A238814 Zhi-Wei Sun, <a href="/A238814/b238814.txt">Table of n, a(n) for n = 1..10000</a>
%H A238814 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014.
%e A238814 a(1) = 2 since prime(2) - 2 + 1 = 3 - 1 = 2 and prime(3) - 3 + 1 = 5 - 2 = 3 are both prime.
%e A238814 a(2) = 3 since prime(3) - 3 + 1 = 5 - 2 = 3 and prime(5) - 5 + 1 = 11 - 4 = 7 are both prime.
%t A238814 p[k_]:=PrimeQ[Prime[Prime[k]]-Prime[k]+1]
%t A238814 n=0
%t A238814 Do[If[p[k]&&p[k+1],n=n+1;Print[n," ",Prime[k]]],{k,1,914}]
%t A238814 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 *)
%o A238814 (PARI) step(p,k)=k++;while(k--,p=nextprime(p+1)); p
%o A238814 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
%Y A238814 Cf. A000040, A234694, A234695, A238766, A238776.
%K A238814 nonn
%O A238814 1,1
%A A238814 _Zhi-Wei Sun_, Mar 05 2014