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%I #42 Jul 02 2018 22:21:50
%S 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 31,31,41,41,43,43,43,43,43,43,53,53,59,59,59,61,59,61,67,71,71,73,71,
%U 73,79,79,79,83,83,79,83,89,89,89,101,101,103,103,109,103,107,109
%N a(n) is the greatest possible least prime in any partition of prime(n) into three primes; n >= 4.
%C 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 Wikipedia, <a href="http://en.wikipedia.org/wiki/Goldbach's_weak_conjecture">Goldbach's weak conjecture</a>
%e 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 a(18) refers to prime(18) = 61 = 11+19+31 = 13+17+31 = 19+19+23, from which a(18)=19.
%o (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 Cf. A024684, A224534.
%K nonn
%O 4,1
%A _David James Sycamore_, Apr 10 2018
%E More terms from _Michel Marcus_, May 13 2018