  For $k\ge1$, 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.
