A051653 -id:A051653 - cp's OEIS frontend

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

Labos Elemer, Dec 15 1999

nonn

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), ...

			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.

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

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

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

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

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

More terms from Michael De Vlieger, Jun 29 2017

A051649 Primes of the form 2310*p + 1 where p is a prime.

Original entry on oeis.org

4621, 11551, 25411, 43891, 108571, 164011, 168631, 224071, 251791, 261031, 316471, 321091, 348811, 376531, 385771, 459691, 528991, 552091, 607531, 626011, 718411, 723031, 732271, 764611, 801571, 815431, 875491, 995611, 1000231, 1055671, 1064911, 1106491, 1161931

Offset: 1

Labos Elemer

nonn

Generalization of A005385; can be called 2310-safe primes.

A002110(5)*p + 1 = 2310*p + 1 (prime).

			11551 is a term because 11551 = 2310*p + 1 is prime, where p = 5.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A002110, A005384, A005385, A007693, A051653.

Select[Table[2310n+1,{n,Prime[Range[100]]}],PrimeQ] (* Harvey P. Dale, Jan 18 2017 *)

isok(k) = isprime(k) && k % 2310 == 1 && isprime((k-1)/2310); \\ Amiram Eldar, Feb 24 2025

a(n) = 2310 * A051653(n) + 1. - Amiram Eldar, Feb 24 2025

cp's OEIS Frontend

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

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