A163847 Starting from a(1)=13, a(n+1) is the smallest prime > a(n) such that 2*a(n) - a(n+1) is also prime.

13, 19, 31, 43, 67, 73, 79, 97, 127, 151, 163, 199, 241, 271, 313, 349, 367, 397, 421, 433, 457, 541, 619, 631, 643, 673, 727, 811, 853, 877, 967, 997, 1087, 1123, 1129, 1171, 1213, 1297, 1303, 1327, 1423, 1447, 1471, 1483, 1543, 1597, 1627, 1657, 1693

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Aug 05 2009

Keywords

• nonn

Comments

This is: select the 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).

It seems all these primes are in the class 1 (mod 6), that is, in A002476 as opposed to A007528.

Examples

			For a(2), the first candidate is the prime 17=13+4, which is not selected because 13-4=9 is not prime.
The next larger candidate is the prime 13+6=19, which is selected as a(2) because 13-6=7 is also prime.
For a(3) the first candidate is the prime 19+4=23, which is not selected because 19-4=15 is not prime.
The next candidate is the prime 19+10=29, which is not selected because the 19-10=9 is not prime.
The next larger candidate, the prime 19+12=31 is selected as a(3), because 19-12=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. A163846, A222532.

Programs

Maple

A163847 := proc(n) option remember; if n = 1 then 13; else for a from procname(n-1)+2 by 2 do if isprime(a) and isprime( 2*procname(n-1)-a) then RETURN(a) ; fi; od: fi; end:
seq(A163847(n),n=1..80) ; # R. J. Mathar, Aug 29 2009

Mathematica

DeltaPrimePrevNext[n_]:=Module[{d, k1, k2}, k1=n-1; k2=n+1; While[ !PrimeQ[k1] || !PrimeQ[k2], k2++; k1-- ]; d=k2-n]; lst13={}; p=13; Do[If[p-DeltaPrimePrevNext[p]>1, AppendTo[lst13, p]; p=p+DeltaPrimePrevNext[p]],{n, 7!}]; lst13
(* Second program: *)
k=6
n=1
Do[If[m==6, Print[n, " ", 13]]; 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, 6, 1000}] (* Zhi-Wei Sun, Feb 25 2013 *)
sp[p_]:=Module[{p1=NextPrime[p]},While[!PrimeQ[2p-p1],p1=NextPrime[p1]];p1]; NestList[ sp,13,50] (* Harvey P. Dale, Aug 09 2023 *)

PARI

first(n) = { my(res = vector(n)); res[1] = 13; for(x=2, n, forprime(p=res[x-1]+1, , if(ispseudoprime(2*res[x-1] - p), res[x]=p; break()))); res; } \\ Iain Fox, Nov 18 2017

Extensions

Definition and comment rephrased by R. J. Mathar, Aug 29 2009