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A302756 a(n) is the least possible greatest prime in any partition of prime(n) into three primes; n >= 4.

%I A302756 #27 May 27 2025 07:31:10
%S A302756 3,5,5,7,7,11,11,13,13,17,17,17,19,23,23,29,29,31,31,31,31,37,41,37,
%T A302756 41,41,41,43,47,47,53,53,61,61,61,61,61,61,61,71,67,71,71,73,79,83,79,
%U A302756 83,83,83,89,89,97,97,101,97,101,97,103,103,107,107,107,113,127,127
%N A302756 a(n) is the least possible greatest prime in any partition of prime(n) into three primes; n >= 4.
%C A302756 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 proven) 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 as the least possible value of the greatest prime r (with lower prime p not constrained to be A302607(n)). The sequence is not strictly increasing, and although many primes appear repeatedly, some do not appear at all (e.g. 59 is not included).
%H A302756 Michel Marcus, <a href="/A302756/b302756.txt">Table of n, a(n) for n = 4..5000</a>
%H A302756 Wikipedia, <a href="http://en.wikipedia.org/wiki/Goldbach&#39;s_weak_conjecture">Goldbach's weak conjecture</a>
%e A302756 The partition of prime(5)=11 into 3 primes p <= q <= r is 11=3+3+5 and since no smaller value than 5 can be attributed to r, a(5)=5.
%o A302756 (PARI) a(n) = {my(pn = prime(n), res = oo); forprime(p=2, pn, forprime(q=p, pn, forprime(r=q, pn, if (p+q+r == pn, res = min(res, r));););); res;} \\ _Michel Marcus_, May 13 2018
%o A302756 (PARI) first(n) = {n = prime(n + 3); my(strt = vector(n, i, i), t = 0, res = vector(primepi(n) - 3)); forprime(p = 2, n, forprime(q = p, n - p, forprime(r = q, n - p - q, strt[p + q + r] = min(r, strt[p + q + r])))); forprime(p = 7, n, t++; res[t] = strt[p]); res} \\ _David A. Corneth_, May 14 2018
%Y A302756 Cf. A024684, A224534, A302607.
%K A302756 nonn
%O A302756 4,1
%A A302756 _David James Sycamore_, Apr 12 2018