TITLE: Upper bound on length of addition chain
QUESTION [8 upvotes]: An addition chain for $n$ is a finite sequence of integers starting at 1 and ending at $n$, such that each element is a sum of two previous elements. A short addition chain for $n$ can be used, for example, to calculate $x^n$ quickly by multiplying previously-calculated values.
Let $l(n)$ be the length of the shortest addition chain for $n$. I am looking for an upper bound on $l(n)$ of the form:
$$l(n) \leq C \cdot \log_2(n)$$
where $C$ is a constant. 
The Wikipedia page cites the following bound from 1975:
$$l(n) \leq \log_2(n) + \log_2(n)(1+o(1))/\log_2(\log_2(n)) < 2\log_2(n)$$
The OEIS page states several bounds, but without a clear reference:
$$l(n) \leq (5/2) \log(n) = 1.73 \log_2(n)$$
$$l(n) \leq (4/3) \lfloor\log_2(n)\rfloor + 2$$
What is a reference for the best known upper bound of the form $l(n) \leq C \cdot \log_2(n)$ (the smallest value of $C$)?

REPLY [2 votes]: In this paper:
Wattel, E. & Jensen, G. "Efficient calculation of powers in a semigroup"
Stichting Mathematisch Centrum. Zuivere Wiskunde, 1968, 1-18
they prove that  $l(n)/log_2(n)\le1.463$ and takes on this value for $n=71$.
This paper is interesting since it's an earlier usage of the Thurber sliding window method than the Thurber paper. The only reason I know about this paper is that it was referenced by a student doing a term paper and he was given the paper by one of the authors.
I don't understand the interest in a bound like this mind you.