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Given a positive integer $n$ in the standard form $$n=\prod_k p_k^{\alpha_k}$$ and the arithmetic function (investigated by Erdős in this paper) $$A(n)=\sum_k \alpha_k p_k$$ let's define the subset $E$ of positive integers $$E=\Big\{n\in N:A(n)\,|\,n,\;A(n)\lt n\Big\}=\Big\{16,27,30,60,70,72,84,105,150,\dots\Big\}$$ I ask if the density of this subset has ever been studied and, in particular, if it is possible to prove the convergence of the series $$\sum_{n\,\in\,E}\frac 1 n$$ After $10^6$ terms ($n=1507900211$) the sum of the series is $0.367044640179285...$, the growth being extremely slow.

Primes and semiprimes do not belong to the subset $E$, but composites of the forms (and many others) $$2^{2^k}p(p+2^{2^k})$$ (with $p$, $p+2^{2^k}$ primes and $k\ge0$) $$p^{(p-1)q}q^p$$ (with $p$, $q$ primes)

are in $E$.

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    $\begingroup$ A user on math.se has commented on the value calculated out to $10^9$. $\endgroup$
    – Gerry Myerson
    Feb 19, 2022 at 23:48
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    $\begingroup$ Doesn't theorem 2.1 in the paper linked in the question handle the cardinality? $\endgroup$
    – Hhhhhhhhhhh
    Feb 20, 2022 at 5:25
  • $\begingroup$ The values you gave seem to show that the $k$-th smallest element of $E$ is greater than the $k$-th smallest half sum of twin primes. Could this be proven, that would entail the convergence of your series. $\endgroup$
    – Sylvain JULIEN
    Mar 2, 2022 at 15:13
  • $\begingroup$ @SylvainJULIEN nice idea, but this is definitely out of reach, as it would need (or imply) the twin prime conjecture to be proven :) $\endgroup$
    – Wolfgang
    Mar 6, 2022 at 15:54

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