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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. 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 (list; graph; refs; listen; history; text; internal format)
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 * 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

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Last modified December 9 00:32 EST 2019. Contains 329871 sequences. (Running on oeis4.)