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A105414 Numbers p(n) such that p(n)+2 and p(n+7)-2 are both prime numbers, where p(n) is the n-th prime. 1

%I #12 Sep 05 2019 09:30:34

%S 17,71,149,191,431,521,821,1049,1277,1289,1451,1619,1667,1877,1949,

%T 2027,2657,3299,3329,3467,3527,3539,3767,3929,4271,4931,5477,5849,

%U 6131,6659,6701,6779,6827,8537,8819,8999,9419,9719,9929,10037,10091,11069,11117

%N Numbers p(n) such that p(n)+2 and p(n+7)-2 are both prime numbers, where p(n) is the n-th prime.

%C Conjecture: There are an infinite number of primes p(n) such that p(n)-2 and p(n+k)-2 are both prime for all k > 1.

%H Harvey P. Dale, <a href="/A105414/b105414.txt">Table of n, a(n) for n = 1..1000</a>

%e p(8)-2 = 17, p(8+6)-2 = 41, both prime, 17 is in the sequence.

%t For[n = 1, n < 500, n++, If[PrimeQ[Prime[n] + 2], If[PrimeQ[Prime[n + 7] - 2], Print[Prime[n]]]]] (* _Stefan Steinerberger_, Feb 07 2006 *)

%t Select[Prime[Range[1500]],AllTrue[{#+2,Prime[PrimePi[#]+7]-2},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Sep 05 2019 *)

%o (PARI) pnpk(n,m,k) = \ both are prime { local(x,l1,l2,v1,v2); for(x=1,n, v1 = prime(x)+ k; v2 = prime(x+m)+k; if(isprime(v1)&isprime(v2), \ print1(x",") print1(v1",") ) ) }

%K nonn

%O 1,1

%A _Cino Hilliard_, May 02 2005

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Last modified July 22 09:51 EDT 2024. Contains 374490 sequences. (Running on oeis4.)