

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.


6



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
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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 ZhiWei 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 2pq 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

ZhiWei Sun, Table of n, a(n) for n = 1..10000
ZhiWei 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 52=3 is also prime.
The first candidate for a(3) is the prime 7+4=11, which is selected since 74=3 is also prime.
The first candidate for a(4) is the prime 11+2=13, which is not selected since 112=9 is composite.
The second candidate for a(4) is the prime 11+4=17, which is selected since 114=7 is prime.


MATHEMATICA

DeltaPrimePrevNext[n_]:=Module[{d, k1, k2}, k1=n1; k2=n+1; While[ !PrimeQ[k1]  !PrimeQ[k2], k2++; k1 ]; d=k2n]; lst={}; p=5; Do[If[pDeltaPrimePrevNext[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}] (* ZhiWei Sun, Feb 25 2013 *)
np[n_]:=Module[{nxt=NextPrime[n]}, While[!PrimeQ[2nnxt], nxt=NextPrime[nxt]]; nxt]; NestList[np, 5, 60] (* Harvey P. Dale, Feb 28 2013 *)


CROSSREFS

Cf. A163847, A222532.
Sequence in context: A059786 A300097 A166574 * A156104 A191080 A293200
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



