OFFSET
1,1
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.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..45 from Jean-Christophe Hervé)
EXAMPLE
a(1) = 11 because 11, (11-1)/2 = 5, (11-2)/3 = 3 and (11-3)/4 = 2 are all primes.
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Jean-Christophe Hervé, Sep 14 2014
STATUS
approved