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A163846 Starting from a(1)=5, a(n+1) is the smallest prime > a(n) such that 2*a(n)-a(n+1) is also prime.

%I A163846 #11 Feb 28 2013 10:32:15
%S A163846 5,7,11,17,23,29,41,53,59,71,83,107,113,137,167,197,227,257,263,269,
%T A163846 281,293,317,353,359,401,419,449,467,491,503,557,593,599,641,683,719,
%U A163846 761,821,881,941,953,977,983,1013,1049,1151,1193,1223,1229,1277,1361,1433
%N A163846 Starting from a(1)=5, a(n+1) is the smallest prime > a(n) such that 2*a(n)-a(n+1) is also prime.
%C A163846 This is: select the smallest prime a(n+1) = a(n)+d such that at a(n)-d is another prime at the same distance to but at the opposite side of a(n).
%C A163846 From _Zhi-Wei Sun_, Feb 25 2013: (Start)
%C A163846 By induction, a(n)==2 (mod 3) for all n>2.
%C A163846 For a prime p>3 define g(p) as the least prime q>p such that 2p-q is also prime. Construct a simple (undirected) graph G as follows: The vertex set is the set of all primes greater than 3, and there is an edge connecting the vertices p and q>p if and only if g(p)=q.
%C A163846 Conjecture: The graph G constructed above consists of exactly two trees: one containing 7 and all odd primes congruent to 2 modulo 3, and another one containing all primes congruent to 1 modulo 3 except 7. (End)
%H A163846 Zhi-Wei Sun, <a href="/A163846/b163846.txt">Table of n, a(n) for n = 1..10000</a>
%H A163846 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arXiv:1211.1588.
%e A163846 The first candidate for a(2) is the prime 5+2=7, which is selected since 5-2=3 is also prime.
%e A163846 The first candidate for a(3) is the prime 7+4=11, which is selected since 7-4=3 is also prime.
%e A163846 The first candidate for a(4) is the prime 11+2=13, which is not selected since 11-2=9 is composite.
%e A163846 The second candidate for a(4) is the prime 11+4=17, which is selected since 11-4=7 is prime.
%t A163846 DeltaPrimePrevNext[n_]:=Module[{d, k1, k2}, k1=n-1; k2=n+1; While[ !PrimeQ[k1] || !PrimeQ[k2], k2++; k1-- ]; d=k2-n]; lst={}; p=5; Do[If[p-DeltaPrimePrevNext[p]>1, AppendTo[lst, p]; p=p+DeltaPrimePrevNext[p]], {n,6!}]; lst
%t A163846 k=3
%t A163846 n=1
%t A163846 Do[If[m==3, Print[n, " ", 5]]; If[m==k, n=n+1; Do[If[PrimeQ[2Prime[m]-Prime[j]]==True, k=j; Print[n, " ", Prime[j]]; Goto[aa]], {j, m+1, PrimePi[2Prime[m]]}]]; Label[aa]; Continue, {m, 3, 1000}] (* _Zhi-Wei Sun_, Feb 25 2013 *)
%t A163846 np[n_]:=Module[{nxt=NextPrime[n]},While[!PrimeQ[2n-nxt],nxt=NextPrime[nxt]]; nxt]; NestList[np, 5, 60] (* _Harvey P. Dale_, Feb 28 2013 *)
%Y A163846 Cf. A163847, A222532.
%K A163846 nonn
%O A163846 1,1
%A A163846 _Vladimir Joseph Stephan Orlovsky_, Aug 05 2009
%E A163846 Edited by _R. J. Mathar_, Aug 29 2009