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

1,1

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)

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.

Example

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.

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 *)

Crossrefs

Cf. A163847, A222532.

Sequence in context: A059786 A300097 A166574 * A354357 A156104 A191080

Adjacent sequences: A163843 A163844 A163845 * A163847 A163848 A163849

Keyword

nonn

Author

Vladimir Joseph Stephan Orlovsky, Aug 05 2009

Extensions

Edited by R. J. Mathar, Aug 29 2009

Status

approved