We present three algorithms to compute the complexity ∥n∥ of all
natural numbers n≤N. The first of them is a brute force algorithm,
computing all these complexities in time O(N2) and space O(Nlog2N). The
main problem of this algorithm is the time needed for the computation. In 2008
there appeared three independent solutions to this problem: V. V. Srinivas and
B. R. Shankar [11], M. N. Fuller [7], and J. Arias de Reyna and J. van de Lune
[3]. All three are very similar. Only [11] gives an estimation of the
performance of its algorithm, proving that the algorithm computes the
complexities in time O(N1+β), where 1+β=log3/log2≈1.584963. The other two algorithms, presented in [7] and
[3], were very similar but both superior to the one in [11]. In Section 2 we
present a version of these algorithms and in Section 4 it is shown that they
run in time O(Nα) and space O(NloglogN). (Here α=1.230175).
In Section 2 we present the algorithm of [7] and [3]. The main advantage of
this algorithm with respect to that in [11] is the definition of kMax in
Section 2.7. This explains the difference in performance from O(N1+β)
to O(Nα).
In Section 3 we present a detailed description a space-improved algorithm of
Fuller and in Section 5 we prove that it runs in time O(Nα) and space
O(N(1+β)/2loglogN), where α=1.230175 and
(1+β)/2≈0.792481.Comment: 21 pages. v2: We improved the computations to get a better bound for
$\alpha