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.

5, 7, 11, 17, 23, 29, 41, 53, 59, 71, 83, 107, 113, 137, 167, 197, 227, 257, 263, 269, 281, 293, 317, 353, 359, 401, 419, 449, 467, 491, 503, 557, 593, 599, 641, 683, 719, 761, 821, 881, 941, 953, 977, 983, 1013, 1049, 1151, 1193, 1223, 1229, 1277, 1361, 1433

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Aug 05 2009

Keywords

• nonn

Comments

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).

From Zhi-Wei Sun, Feb 25 2013: (Start)

By induction, a(n)==2 (mod 3) for all n>2.

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.

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)

Examples

			The first candidate for a(2) is the prime 5+2=7, which is selected since 5-2=3 is also prime.
The first candidate for a(3) is the prime 7+4=11, which is selected since 7-4=3 is also prime.
The first candidate for a(4) is the prime 11+2=13, which is not selected since 11-2=9 is composite.
The second candidate for a(4) is the prime 11+4=17, which is selected since 11-4=7 is prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Crossrefs

Cf. A163847, A222532.

Programs

Mathematica

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
k=3
n=1
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 *)
np[n_]:=Module[{nxt=NextPrime[n]},While[!PrimeQ[2n-nxt],nxt=NextPrime[nxt]]; nxt]; NestList[np, 5, 60] (* Harvey P. Dale, Feb 28 2013 *)

Extensions

Edited by R. J. Mathar, Aug 29 2009