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

%I A247347 #30 Mar 09 2024 17:55:06
%S A247347 11,23,719,1439,5639,25799,28319,35879,56039,58679,77279,98999,104759,
%T A247347 121559,166919,174599,206639,253679,334319,350159,424079,433439,
%U A247347 451679,452759,535919,539159,582719,595319,645839,671039,743279,818999,824039
%N A247347 Primes p such that (p-k)/(k+1) is also prime for k = 1, 2, 3.
%C A247347 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.
%C A247347 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).
%C A247347 These numbers generate sequences 4-3-2-1 in A052126.
%C A247347 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.
%H A247347 Amiram Eldar, <a href="/A247347/b247347.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..45 from Jean-Christophe Hervé)
%e A247347 a(1) = 11 because 11, (11-1)/2 = 5, (11-2)/3 = 3 and (11-3)/4 = 2 are all primes.
%t A247347 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
%t A247347 Select[Prime[Range[70000]],AllTrue[Table[(#-k)/(k+1),{k,3}],PrimeQ]&] (* _Harvey P. Dale_, Mar 09 2024 *)
%o A247347 (PARI) isokp(v) = (type(v) == "t_INT") && isprime(v);
%o A247347 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
%Y A247347 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).
%K A247347 nonn
%O A247347 1,1
%A A247347 _Jean-Christophe Hervé_, Sep 14 2014