In a recent video (https://www.facebook.com/188916357807416/videos/519169035700435/) Stephen Wolfram wonders whether, for every integer n>2, eventually the number of integers which are precisely the product of n primes (not necessarily different) is greater than the number of integers which are precisely the product of n-1 primes. For n = 2, as he shows, this seems to be the case beyond somewhere around 10,000. I am certain this has been studied and settled (Paul Erdös perhaps?), but don´t know where or by whom.
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asked Apr 15, 2020 at 21:49
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3$\begingroup$ This is true for all $n\geq 1$. Asymptotics are known and are due to Landau, more details here $\endgroup$– WojowuCommented Apr 15, 2020 at 21:55
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4$\begingroup$ The main terms of these asymptotics suggests that the point at which $k+1$-almost primes overtake $k$-almost primes is around $e^{e^k}$. $\endgroup$– WojowuCommented Apr 15, 2020 at 22:04
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