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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.
See also Brauer Chain, Hansen Number
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.