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A105413 Primes p = prime(k) such that both p+2 and prime(k+6)-2 are prime numbers.

%I A105413 #35 Oct 04 2024 06:56:25
%S A105413 3,11,107,239,311,569,1019,1031,1229,1427,1997,2081,2087,2111,2687,
%T A105413 3251,4049,4127,4157,4229,4241,4481,5231,5639,6089,7307,7559,8969,
%U A105413 9629,10007,10457,13691,13829,13901,14249,14549,14561,16187,16649,17207
%N A105413 Primes p = prime(k) such that both p+2 and prime(k+6)-2 are prime numbers.
%C A105413 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 A105413 Amiram Eldar, <a href="/A105413/b105413.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Harvey P. Dale)
%e A105413 prime(5) = 11, and both prime(5)+2 = 13 and prime(5+6)-2 = 29 are primes, so 11 is in the sequence.
%t A105413 For[n = 1, n < 500, n++, If[PrimeQ[Prime[n] + 2], If[PrimeQ[Prime[n + 6] - 2], Print[Prime[n]]]]] (* _Stefan Steinerberger_, Feb 07 2006 *)
%t A105413 Transpose[Select[Partition[Prime[Range[2000]],7,1],#[[2]]-#[[1]] == #[[7]]- #[[6]] == 2&]][[1]] (* _Harvey P. Dale_, Oct 08 2014 *)
%o A105413 (PARI) pnpk(n, m=6, 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 A105413 (PARI) lista(pmax) = {my(k = 1, p = primes(7)); forprime(p1 = p[#p], pmax, k++; p[#p] = p1; if(p[2]- p[1] == 2 && p[7] - p[6] == 2, print1(p[1], ", ")); for(i = 1, #p-1, p[i] = p[i+1]));} \\ _Amiram Eldar_, Oct 04 2024
%o A105413 (Magma) [NthPrime(n): n in [1..2000] | IsPrime(NthPrime(n)+2) and IsPrime(NthPrime(n+6)-2)]; // _Vincenzo Librandi_, Sep 14 2015
%Y A105413 Cf. A089635, A105409, A105410, A105411, A105412, A105414.
%K A105413 nonn
%O A105413 1,1
%A A105413 _Cino Hilliard_, May 02 2005