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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.
0
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
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
Sequence in context: A064915 A375969 A180648 * A175740 A320321 A215009
KEYWORD
nonn,more
AUTHOR
Frank M Jackson, Dec 20 2018
STATUS
approved