A105412 - OEIS

%I #32 Oct 04 2024 06:56:22

%S 5,41,179,197,281,599,641,809,827,857,1061,1451,2237,2549,3119,3329,

%T 3359,3821,4001,4091,4217,5417,5441,5849,6269,6659,6761,6791,7457,

%U 7949,8387,8597,9239,9419,9431,9677,10301,10427,10859,10889,11117,11717

%N Primes p = prime(k) such that both p+2 and prime(k+5)-2 are prime numbers.

%C Conjecture: There are infinitely many primes p(k) such that p(k)+2 and p(k+m)-2 are both primes for all m > 1.

%H Amiram Eldar, <a href="/A105412/b105412.txt">Table of n, a(n) for n = 1..10000</a>

%e prime(13) = 41, and both prime(13)+2 = 43 and prime(13+5)-2 = 59 are primes, so 41 is in the sequence.

%t For[n = 1, n < 500, n++, If[PrimeQ[Prime[n] + 2], If[PrimeQ[Prime[n + 5] - 2], Print[Prime[n]]]]] (* _Stefan Steinerberger_, Feb 07 2006 *)

%o (PARI) pnpk(n, m=5, k=2) = { local(x, v1, v2); for(x=1, n, v1 = prime(x)+ k; v2 = prime(x+m)-k; if(isprime(v1)&isprime(v2), print1(prime(x), ", ") ) ) ;} \\ corrected by _Michel Marcus_, Sep 14 2015

%o (PARI) lista(pmax) = {my(k = 1, p = primes(6)); forprime(p1 = p[#p], pmax, k++; p[#p] = p1; if(p[2]- p[1] == 2 && p[6] - p[5] == 2, print1(p[1], ", ")); for(i = 1, #p-1, p[i] = p[i+1]));} \\ _Amiram Eldar_, Oct 04 2024

%o (Magma) [NthPrime(n): n in [1..1500] | IsPrime(NthPrime(n)+2) and IsPrime(NthPrime(n+5)-2)]; // _Vincenzo Librandi_, Sep 14 2015

%Y Cf. A105409, A105410, A105411, A105413, A105414.

%K nonn

%O 1,1

%A _Cino Hilliard_, May 02 2005