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%I #46 Dec 22 2018 16:27:38
%S 5,7,10,16,36,210
%N n such that all n - p are prime where p is prime in range n/2 <= p < n.
%C The following is a quotation from Hage-Hassan in his paper (see Link below): "The (concept of) right and left symmetry is fundamental in physics. This incites us to ask whether this symmetry is in (the) primes. Find the numbers n with a + a' = n. a, a' are primes and {a} are all the primes with: n/2 <= a < n and n = 2,3, ...".
%C By conjecture there are only 6 terms and this has been checked for n up to 10^8. Limiting the sequence to even numbers, by conjecture, the 4 numbers 10, 16, 36, 210 are the only even numbers that when partitioned to be the sum of two primes per the Goldbach conjecture use all primes p in the range n/2 <= p < n to provide solutions.
%H Mehdi Hage-Hassan, <a href="https://hal.archives-ouvertes.fr/hal-00879586/document">An elementary introduction to Quantum mechanic</a>, hal-00879586 2013 pp 58.
%e a(5)=36, because the primes in the range 18 <= p < 36 are {19, 23, 29, 31}. Also the complementary set {17, 13, 7, 5} has all its members prime. This is the 5th occurrence of such a number.
%t plst[n_] := Select[Range[Ceiling[n/2], n-1], PrimeQ]; lst={}; Do[If[plst[n]!={}&&AllTrue[n-plst[n], PrimeQ], AppendTo[lst, n]], {n, 1, 1000}]; lst
%K nonn,more
%O 1,1
%A _Frank M Jackson_, Dec 18 2018
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