Almost primes and the Banks-Martin conjecture

It has been known since Erdos 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 $k=6$.

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