乘法表问题的算法
Algorithms for the Multiplication Table Problem
摘要：
令$M(n)$表示$n \times n$乘法表中不同条目的数量。函数$M(n)$已经被Erdens、Tenenbaum、Ford和其他人研究过，但是$M(n)$作为$n \to \infty$的渐近行为还不清楚。因此，人们对精确地或近似地计算$M(n)$的算法感兴趣。比较了几种精确计算$M(n)$的算法，给出了一种新的算法，该算法具有次二次运行时间。我们还提出了两个Monte Carlo算法的近似计算$M(n)$。我们给出了$n$到$2^{30}$的精确计算结果，以及$n$到$2^{100,000,000}$的Monte Carlo计算结果，并将我们的实验结果与Ford的数量级结果进行了比较。
英文摘要：
Let $M(n)$ denote the number of distinct entries in the $n \times n$ multiplication table. The function $M(n)$ has been studied by Erdős, Tenenbaum, Ford, and others, but the asymptotic behaviour of $M(n)$ as $n \to \infty$ is not known precisely. Thus, there is some interest in algorithms for computing $M(n)$ either exactly or approximately. We compare several algorithms for computing $M(n)$ exactly, and give a new algorithm that has a subquadratic running time. We also present two Monte Carlo algorithms for approximate computation of $M(n)$. We give the results of exact computations for values of $n$ up to $2^{30}$, and of Monte Carlo computations for $n$ up to $2^{100,000,000}$, and compare our experimental results with Ford's order-of-magnitude result.
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