A067027 Numbers n such that (prime(n)# + 4)/2 is a prime, where x# is the primorial A034386(x).

1, 2, 3, 4, 6, 10, 11, 12, 15, 17, 29, 48, 63, 77, 88, 187, 190, 338, 1133, 1311, 1832, 2782, 2907, 3180, 3272, 5398, 17530

Offset: 1

Author

Labos Elemer, Dec 29 2001

Keywords

• nonn

Comments

Numbers n such that [A002110(n)/2]+2 is prime.

These primes are products of consecutive odd primes plus 2: 2+[3.5.7.....p(n)] if n is here.

a(19)-a(22) are Fermat and Lucas PRPs. (prime(2782)# + 4)/2 has 10865 digits. PFGW Version 1.2.0 for Windows [FFT v23.8] Primality testing (p(2782)#+4)/2 [N-1/N+1, Brillhart-Lehmer-Selfridge] Running N-1 test using base 5 Running N+1 test using discriminant 13, base 1+sqrt(13) (p(2782)#+4)/2 is Fermat and Lucas PRP! - Jason Earls, Dec 12 2006

a(28) > 25000. - Robert Price, Sep 29 2017

Crossrefs

Cf. A002110, A067024, A065026.

Programs

Mathematica

p = 1; Do[p = p*Prime[n]; If[PrimeQ[(p + 4)/2], Print[n]], {n, 1, 400} ]
Flatten[Position[FoldList[Times,Prime[Range[3000]]],?(PrimeQ[ (#+4)/2]&)]] (* _Harvey P. Dale, May 24 2015 *)

PARI

n=0;pr=1/2;forprime(p=2,1e4,n++;pr*=p;if(ispseudoprime(pr+2),print1(n", "))) \\ Charles R Greathouse IV, Jul 25 2011

Extensions

More terms from Robert G. Wilson v, Dec 30 2001

a(19)-a(22) from Jason Earls, Dec 12 2006

a(23) from Ray Chandler, Jun 16 2013

a(24)-a(27) from Robert Price, Sep 29 2017