A320447 n such that all n - p are prime where p is prime in range n/2 <= p < n.

5, 7, 10, 16, 36, 210

Offset: 1

Author

Frank M Jackson, Dec 18 2018

Keywords

• nonn
• more

Comments

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, ...".

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.

Examples

			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.
		

Links

• Mehdi Hage-Hassan, An elementary introduction to Quantum mechanic, hal-00879586 2013 pp 58.

Programs

Mathematica

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