A247348 Primes p such that (p-k)/(k+1) is also prime for k = 1, 2, 3, 4.

174599, 334319, 535919, 671039, 907199, 2129399, 2298119, 3103799, 3369959, 4351199, 4598159, 5697599, 6184799, 6446159, 7224839, 7943759, 7957319, 8148839, 8346959, 8656919, 9096359, 9339119, 9463319, 9511199, 10514159, 10780559, 11816999, 12424319, 13781039

Offset: 1

Author

Jean-Christophe Hervé, Sep 14 2014

Keywords

• nonn

Comments

Could be called 4-safe primes, or safe primes of order 4, as the safe primes are the primes such that (p-1)/2 is prime.

Obviously a subsequence of the k-safe primes for k < 4 : A005385 (safe primes, k=1), A181841 (supersafe primes, k=2), A247347 (k=3).

a(n) = 119 (mod 120) for all n.

These numbers generate sequences 5-4-3-2-1 in A052126.

Links

• Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A005385 (safe primes), A181841 (supersafe primes), A247347 (3-safe primes), A163573 (similar definition with (p+k)/(k+1) as primes).

Programs

Mathematica

lst={}; Do[p=Prime[n]; If[PrimeQ[(p-1)/2]&&PrimeQ[(p-2)/3]&&PrimeQ[(p-3)/4]&&PrimeQ[(p-4)/5], AppendTo[lst, p]], {n, 2*9!}]; lst
Select[Prime[Range[900000]],AllTrue[Table[(#-k)/(k+1),{k,4}],PrimeQ]&] (* Harvey P. Dale, Jul 07 2025 *)

PARI

isokp(v) = (type(v) == "t_INT") && isprime(v);
lista(nn) = {forprime(p=2, nn, if (isokp((p-1)/2) && isokp((p-2)/3) && isokp((p-3)/4) && isokp((p-4)/5), print1(p, ", ")););} \\ Michel Marcus, Sep 15 2014

Python

from _future_ import division
from sympy import prime, isprime
A247348_list = [p for p in (5*prime(n)+4 for n in range(1,10**6)) if not ((p-1) % 2 or (p-2) % 3 or (p-3) % 4) and isprime(p) and isprime((p-1)//2) and isprime((p-2)//3) and isprime((p-3)//4)] # Chai Wah Wu, Sep 18 2014

Extensions

More terms from Michel Marcus, Sep 15 2014