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 A062391 #21 Sep 14 2016 16:56:08
%S A062391 3,5,11,13,17,23,31,43,53,61,67,71,73,79,89,101,103,107,127,139,167,
%T A062391 173,181,193,197,211,223,227,233,241,269,277,281,349,353,359,379,433,
%U A062391 467,499,521,523,557,577,587,613,631,743,757,769,821,827,829,883,947
%N A062391 a(1) = 3, a(2) = 5; a(n+1) = smallest prime number > a(n) such that the sum of any three consecutive terms is a prime.
%C A062391 What is the longest string of consecutive primes? A derived sequence could be the start of the first occurrence of a string of n consecutive primes in this sequence.
%C A062391 See A072225 for relevant info and links. - _Zak Seidov_, Sep 14 2016
%H A062391 Zak Seidov, <a href="/A062391/b062391.txt">Table of n, a(n) for n = 1..1000.</a>
%e A062391 After 43, the next term is 53, since 31 + 43 + 47 = 121 is not prime and 31 + 43 + 53 = 127 is prime.
%t A062391 a=3; b=5; lst={a, b}; Do[c=a+b+n; If[PrimeQ[c]&&n>b&&PrimeQ[n], AppendTo[lst, n]; a=b; b=n], {n, 0, 8!}]; lst (* _Vladimir Joseph Stephan Orlovsky_, Dec 17 2008 *)
%t A062391 nxt[{a_,b_}]:=Module[{p=NextPrime[b]},While[!PrimeQ[a+b+p],p= NextPrime[ p]];{b,p}]; Transpose[NestList[nxt,{3,5},70]][[1]] (* _Harvey P. Dale_, Aug 05 2013 *)
%o A062391 (PARI) { n=a1=0; forprime (p=3, 5*10^5, if (p<6 || isprime(p + s), write("b062391.txt", n++, " ", p); s=a1 + p; a1=p; if (n==1000, break)) ) } \\ _Harry J. Smith_, Aug 07 2009
%Y A062391 Cf. A072536, A072537, A000040.
%Y A062391 Cf. A072225.
%K A062391 nonn,easy
%O A062391 1,1
%A A062391 _Amarnath Murthy_, Jun 27 2001
%E A062391 Corrected and extended by Larry Reeves (larryr(AT)acm.org), Jul 02 2001