A247347 Primes p such that (p-k)/(k+1) is also prime for k = 1, 2, 3.

11, 23, 719, 1439, 5639, 25799, 28319, 35879, 56039, 58679, 77279, 98999, 104759, 121559, 166919, 174599, 206639, 253679, 334319, 350159, 424079, 433439, 451679, 452759, 535919, 539159, 582719, 595319, 645839, 671039, 743279, 818999, 824039

Offset: 1

Author

Jean-Christophe Hervé, Sep 14 2014

Keywords

• nonn

Comments

Could be called 3-safe primes, or safe primes of order 3, as the safe primes are the primes such that (p-1)/2 is prime.

Obviously a subsequence of the safe primes A005385 and of the supersafe primes A181841; thus (a(n)-1)/2 is a Sophie Germain prime (cf. A005384).

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

a(n) == -1 (mod 120) for n > 2: because (a(n)-1)/2, (a(n)-2)/3 and (a(n)-3)/4 must be integer, a(n) = -1 (mod 12), thus a(n) = -1 (mod 24) or a(n) = 11 mod(24) for all n; if a(n) = 11 (mod 24), (a(n)-3)/4 = 2 (mod 24) and would be even and not prime unless n=1; thus a(n) = -1 (mod 24) for n > 1. Now, if a(n) = 23 or 47 or 71 or 95 (mod 120), one of the (a(n)-k)/k is a multiple of 5 and thus not prime unless n = 2 and a(2) = 23 (in which case (23-3)/4 is exactly 5); thus a(n) == -1 (mod 120) for n > 2.

Examples

			a(1) = 11 because 11, (11-1)/2 = 5, (11-2)/3 = 3 and (11-3)/4 = 2 are all primes.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..45 from Jean-Christophe Hervé)

Crossrefs

Cf. A005384 (Sophie Germain primes), A005385 (safe primes), A181841 (supersafe primes), A247348 (4-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], AppendTo[lst, p]], {n, 2*9!}]; lst
Select[Prime[Range[70000]],AllTrue[Table[(#-k)/(k+1),{k,3}],PrimeQ]&] (* Harvey P. Dale, Mar 09 2024 *)

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), print1(p, ", ")););} \\ Michel Marcus, Sep 15 2014