A number 
for which a shortest chain exists which is a Brauer chain
is called a Brauer number. There are infinitely many non-Brauer numbers.
Brauer Number
See also
Brauer Chain, Hansen NumberExplore with Wolfram|Alpha
References
Guy, R. K. "Addition Chains. Brauer Chains. Hansen Chains." §C6 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 111-113, 1994.Referenced on Wolfram|Alpha
Brauer NumberCite this as:
Weisstein, Eric W. "Brauer Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BrauerNumber.html