On Erdős sums of almost primes - Gorodetsky, O; Lichtman, JD; Wong, MD (Elsevier, 2023)
In 1935, Erdős proved that the sums fk = Pn1/(n log n), over integers n with exactly k prime factors, are bounded by an absolute constant, and in 1993 Zhang proved that fk is maximized by the prime sum f1 = Pp1/(p log p). According to a 2013 conjecture of Banks and Martin, the sums fk are predicted to decrease monotonically in k. In this article, we show that the sums restricted to odd integers are indeed monotonically decreasing in k, sufficiently large. By contrast, contrary to the conjecture we prove that the sums fk increase monotonically in k, sufficiently large.