A051887 Minimal and special 2k-Germain primes, where 2k is in A002110 (primorial numbers).

2, 2, 2, 2, 2, 5, 17, 11, 11, 11, 2, 23, 7, 43, 19, 3, 5, 2, 7, 3, 61, 53, 2, 41, 47, 2, 5, 7, 31, 2, 47, 13, 113, 7, 137, 103, 43, 41, 97, 3, 173, 97, 41, 13, 97, 59, 29, 53, 3, 107, 127, 197, 3, 487, 433, 31, 281, 587, 7, 89, 41, 47, 193, 239, 41, 7, 31, 67

Offset: 1

Author

Labos Elemer, Dec 15 1999

Keywords

• nonn

Comments

a(n) is the minimal prime p such that primorial(n)*p + 1 is also prime.

While p is in A005384, the primorial(n)*p + 1 primes are in A051902 (primorial-safe primes).

Analogous to or subset of A051686, where the even numbers are 2, 6, ..., A002110(n), ...

Examples

			a(25) = 47 because primorial(25)*47 + 1 is also prime and minimal with this property: primorial(25)*47 + 1 = 47*2305567963945518424753102147331756070 + 1 = 108361694305439365963395800924592535291 is a minimal prime. The first 6 terms (2,2,2,2,2,5) correspond to first entries in A005384, A007693, A051645, A051647, A051653, A051654, respectively.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A002110, A005384, A005385, A051686, A007693, A051886, A051888, A051902.

Programs

Mathematica

Table[p = 2; While[! PrimeQ[Product[Prime@ i, {i, n}] p + 1], p = NextPrime@ p]; p, {n, 68}] (* Michael De Vlieger, Jun 29 2017 *)

PARI

a(n) = {my(p = 2, r = vecprod(primes(n))); while(!isprime(p * r + 1), p = nextprime(p+1)); p;} \\ Amiram Eldar, Feb 25 2025

Formula

a(n) = (A051902(n)-1)/A002110(n). - Amiram Eldar, Feb 25 2025

Extensions

More terms from Michael De Vlieger, Jun 29 2017