Abstract It has been known since Erdős that the sum of 1 / ( n log ⁡ n ) over numbers n with exactly k prime factors (with repetition) is bounded as k varies. We prove that as k tends to infinity, this sum tends to 1. Banks and Martin have conjectured that these sums decrease monotonically in k, and in earlier papers this has been shown to hold for k up to 3. However, we show that the conjecture is false in general, and in fact a global minimum occurs at k = 6 .

中文翻译：

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几乎素数和班克斯-马丁猜想

摘要 自从 Erdős 以来就知道 1 / ( n log ⁡ n ) 在具有恰好 k 个素因子（有重复）的数 n 上的总和随着 k 的变化而有界。我们证明，当 k 趋于无穷大时，这个总和趋于 1。Banks 和 Martin 推测这些总和在 k 中单调递减，并且在早期的论文中，这已经证明当 k 达到 3 时也成立。然而，我们证明了猜想通常是错误的，实际上全局最小值出现在 k = 6 处。