This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A105414 #21 Oct 04 2024 06:56:29
%S A105414 17,71,149,191,431,521,821,1049,1277,1289,1451,1619,1667,1877,1949,
%T A105414 2027,2657,3299,3329,3467,3527,3539,3767,3929,4271,4931,5477,5849,
%U A105414 6131,6659,6701,6779,6827,8537,8819,8999,9419,9719,9929,10037,10091,11069,11117
%N A105414 Primes p = prime(k) such that p+2 and prime(k+7)-2 are both prime numbers.
%C A105414 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 A105414 Amiram Eldar, <a href="/A105414/b105414.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Harvey P. Dale)
%e A105414 p(8)-2 = 17, p(8+6)-2 = 41, both prime, 17 is in the sequence.
%t A105414 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 A105414 Select[Prime[Range[1500]],AllTrue[{#+2,Prime[PrimePi[#]+7]-2},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Sep 05 2019 *)
%o A105414 (PARI) pnpk(n, m=7, 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(v1-k, ", ") ) ) ; } \\ corrected by _Amiram Eldar_, Oct 04 2024
%o A105414 (PARI) lista(pmax) = {my(k = 1, p = primes(8)); forprime(p1 = p[#p], pmax, k++; p[#p] = p1; if(p[2]- p[1] == 2 && p[8] - p[7] == 2, print1(p[1], ", ")); for(i = 1, #p-1, p[i] = p[i+1]));} \\ _Amiram Eldar_, Oct 04 2024
%Y A105414 Cf. A105409, A105410, A105411, A105412, A105413.
%K A105414 nonn
%O A105414 1,1
%A A105414 _Cino Hilliard_, May 02 2005