An addition chain for n is defined to be a sequence (a0β,a1β,β¦,arβ)
such that a0β=1, arβ=n, and, for any 1β€kβ€r, there exist 0β€i,j<k such that akβ=aiβ+ajβ; the number r is called the length of the
addition chain. The shortest length among addition chains for n, called the
addition chain length of n, is denoted β(n). The number β(n) is
always at least log2βn; in this paper we consider the difference
Ξ΄β(n):=β(n)βlog2βn, which we call the addition chain defect.
First we use this notion to show that for any n, there exists K such that
for any kβ₯K, we have β(2kn)=β(2Kn)+(kβK). The main result is
that the set of values of Ξ΄β is a well-ordered subset of
[0,β), with order type ΟΟ. The results obtained here are
analogous to the results for integer complexity obtained in [1] and [3]. We
also prove similar well-ordering results for restricted forms of addition chain
length, such as star chain length and Hansen chain length.Comment: 19 page