Halton sequence

Description

Let $b \ge 2$ be an integer. Each integer $n \ge 0$ has a unique base-$b$ expansion $n = \sum_{k \ge 0} a_k b^k$ with digits $a_k \in \set{0, \ldots, b-1}$, almost all zero. The radical inverse in base $b$ is $$\varphi_b(n) = \sum_{k \ge 0} a_k \, b^{-k-1} \in [0,1).$$ It mirrors the digits of $n$ at the radix point. The sequence $(\varphi_b(n))_{n \ge 0}$ is the van der Corput sequence in base $b$. Let $s \ge 1$ and let $b_1, \ldots, b_s \ge 2$ be pairwise coprime integers. The Halton sequence in these bases is the sequence of points $$x_n = \big(\varphi_{b_1}(n), \ldots, \varphi_{b_s}(n)\big) \in [0,1)^s, \qquad n = 0, 1, 2, \ldots$$ The usual choice is $b_1, \ldots, b_s$ the first $s$ prime numbers.