For $k\ge 1$, a k-almost prime is a positive integer with exactly k prime factors, counted with multiplicity. In this article we give elementary proofs of precise asymptotics for the reciprocal sum of k-almost primes. Our results match the strength of those of classical analytic methods. We also study the limiting behavior of the constants appearing in these estimates, which may be viewed as higher analogues of the Mertens constant $\beta =0.2614...$ Further, in the case $k=2$ of semiprimes we give yet finer-scale and explicit estimates, as well as a conjecture.

中文翻译：

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几乎素数的更高 Mertens 常数

为了 $ķ\ge 1$，一个k几乎素数是一个正整数，正好有k个素因数，以多重性计数。在这篇文章中，我们给出了k几乎素数的倒数和的精确渐近的基本证明。我们的结果与经典分析方法的强度相匹配。我们还研究了这些估计中出现的常数的限制行为，这可能被视为 Mertens 常数的更高类似物$\beta =0.2614...$ 此外，在这种情况下 $ķ=2$ 对于半素数，我们给出了更精细和更明确的估计，以及一个猜想。