# On the density of a particular subset of integers

Given a positive integer $$n$$ in the standard form $$n=\prod_k p_k^{\alpha_k}$$ and the arithmetic function $$f(n)=\sum_k \alpha_k p_k$$ let's define the subset $$F$$ of positive integers $$F=\Big\{n\in N:f(n)\,|\,n,\;f(n)\lt n\Big\}=\Big\{16,27,30,60,70,72,84,105,150,\dots\Big\}$$ 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 $$\sum_{n\,\in\,F}\frac 1 n$$ Numerical experiments would show the convergence of such series towards a value quite close to the inverse of Euler's number $$\sum_{n\,\in\,F}\frac 1 n\sim\frac 1 e$$

Edit

My script is still running, but after $$5\cdot 10^5$$ terms ($$n=584504910$$) the sum of the series is $$0.36652132586744884...\;(\frac 1 e = 0,36787944117144232...)$$: the growth is extremely slow.

Work in progress

The most recent values obtained are the following:

$$n=9928531324,\;\;3986000$$-th term of the series$$,\;\;$$partial sum$$\,=0.36776500537719703...$$

$$n=9931911561,\;\;3987000$$-th term of the series$$,\;\;$$partial sum$$\,=0.36776510608002266...$$

$$n=9935361024,\;\;3988000$$-th term of the series$$,\;\;$$partial sum$$\,=0.36776520674763440...$$

$$n=9938801814\,(\sim 10^{10}),\;\;3989000$$-th term of the series$$,\;\;$$partial sum$$\,=0.36776530738064540...$$

I am cautiously optimistic about the convergence of the series.

• Looks like this is A046346 on OEIS, except for the 4. (70 is in your set, right?) Feb 17 at 7:24
• Many thanks for your suggestion. I have corrected the set. Feb 17 at 7:32
• For every prime pair $p, p+2$, we have $2p(p+2) \in F$. Not sure if this suggests a difficulty related to the prime pair conjecture, or if it's not that important since there are other elements too. Feb 17 at 15:12
• With the numbers with the desired property upto $2.9\cdot 10^9$ I got the sum $$0.36733612496219974942232403795926400423$$ but my guess would still be that the sum slowly diverges. Feb 17 at 17:10
• Nobody still hasn't verified this question??? If the sum converges, then the OP should publish it. No matter it converges to $\frac{1}{e}$ or not. This was one of the most interesting question I've seen in this site! Mar 12 at 16:23