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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 (list; graph; refs; listen; history; text; internal format)



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.


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

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


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.


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


Cf. A002858, A320447.

Sequence in context: A246077 A064915 A180648 * A175740 A320321 A215009

Adjacent sequences:  A322602 A322603 A322604 * A322606 A322607 A322608




Frank M Jackson, Dec 20 2018



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Last modified April 20 22:36 EDT 2021. Contains 343140 sequences. (Running on oeis4.)