%I #15 Jul 12 2023 11:08:01

%S 2,3,4,5,6,7,8,9,10,12,14,17,19,24,29,34,39,44

%N Numbers k such that all k - u are Ulam numbers (A002858) where u is an Ulam number in the range k/2 <= u < k.

%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 This sequence is analogous to A320447. Instead of the sequence of primes it uses the sequence of Ulam numbers (A002858). It is conjectured that the sequence is finite and full.

%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(10)=12, because the Ulam numbers u in the range 6 <= u < 12 are {6, 8, 11}. Also the complementary set {6, 4, 1} has all its members Ulam numbers. This is the 10th occurrence of such a number.

%t Ulam[n_] := Module[{ulams={1, 2}, p}, Do[AppendTo[ulams, p=Last[ulams]; While[p++; Length[DeleteCases[Intersection[ulams, p-ulams], p/2, 1, 1]]!=2]; p], {n-2}]; ulams]; ulst=Ulam[1000]; plst[n_] := Select[ulst, Ceiling[n/2]<=#<n &]; lst={}; Do[If[plst[n]!={}&&Intersection[ulst, nlst=Sort[n-plst[n]]]==nlst, AppendTo[lst, n]], {n, 1, 1000}]; lst

%Y Cf. A002858, A320447.

%K nonn,more

%O 1,1

%A _Frank M Jackson_, Dec 20 2018