A322605 Numbers k such that all k - u are Ulam numbers (A002858) where u is an Ulam number in the range k/2 <= u < k.

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

Offset

1,1

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, ..."

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.

Links

Table of n, a(n) for n=1..18.

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

Example

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.

Mathematica

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

Crossrefs

Cf. A002858, A320447.

Sequence in context: A246077 A064915 A180648 * A175740 A320321 A215009

Adjacent sequences: A322602 A322603 A322604 * A322606 A322607 A322608

Keyword

nonn,more

Author

Frank M Jackson, Dec 20 2018

Status

approved