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 A302607 #42 Jul 02 2018 22:21:50
%S A302607 2,3,3,5,5,5,7,7,11,11,13,13,17,17,19,19,19,19,19,23,29,29,29,31,29,
%T A302607 31,31,41,41,43,43,43,43,43,43,53,53,59,59,59,61,59,61,67,71,71,73,71,
%U A302607 73,79,79,79,83,83,79,83,89,89,89,101,101,103,103,109,103,107,109
%N A302607 a(n) is the greatest possible least prime in any partition of prime(n) into three primes; n >= 4.
%C A302607 Goldbach's weak (ternary) conjecture states that every odd number > 5 can be expressed as the sum of three primes (see link). This sequence applies the conjecture (now proved) to primes > 5. From all possible partitions of prime(n) = p+q+r for primes p,q,r (p <= q <= r), a(n) is chosen to be the maximum possible value of the least prime p. The sequence is not strictly increasing, and although many primes are repeated, some do not appear at all (e.g., 37 and 47 are not included).
%H A302607 Wikipedia, <a href="http://en.wikipedia.org/wiki/Goldbach's_weak_conjecture">Goldbach's weak conjecture</a>
%e A302607 a(4) refers to prime(4) = 7 = 2+2+3 and since there is no (ordered) partition of 7 starting with a greater prime than 2, a(1)=2.
%e A302607 a(18) refers to prime(18) = 61 = 11+19+31 = 13+17+31 = 19+19+23, from which a(18)=19.
%o A302607 (PARI) a(n) = {my(pn = prime(n), res = 0); forprime(p=2, pn, forprime(q=p, pn, forprime(r=q, pn, if (p+q+r == pn, res = max(res, p));););); res;} \\ _Michel Marcus_, May 13 2018
%Y A302607 Cf. A024684, A224534.
%K A302607 nonn
%O A302607 4,1
%A A302607 _David James Sycamore_, Apr 10 2018
%E A302607 More terms from _Michel Marcus_, May 13 2018