In this post we denote the Stirling number of the second kind as ${n\brace k}$, I add as reference the article *Stirling numbers of the second kind* from the encyclopedia Wikipedia. And we denote the sum of divisors function as $\sigma(n)=\sum_{1\leq d\mid n}d$.

I wondered if it is possible to prove some of the following conjectures that I got using a Pari/GP program inspired in an article that refers [1].

**Conjecture 1.** *The estimate* $$\frac{1}{n}\sum_{1\leq k\leq n}\frac{\sigma\left({n\brace k}\right)}{{n\brace k}}>C\cdot\log\log\log n$$
*holds for all* $n\geq 16$, *being* $C>0$ *a suitable constant.*

**Conjecture 2.** *One has that* $$\frac{\sum_{1\leq k\leq n}\sigma\left({n\brace k}\right)}{n^{4+\frac{n}{2}}}\to\infty$$
*as* $n\to\infty$.

Question.Is it possible to prove Conjecture 1 or Conjecture 2?Many thanks.

I think that these conjectures aren't more interesting than studied in [1], but I've curiosity about if it can be deduced (from the few computational evidence that I have, my experiments tell me that these conjectures can aren't sharpest). I would like to know what work can be done to elucidate the veracity of our conjectures.

## References:

[1] F. Luca, *Sums of divisors of binomial coefficients*, Int. Math. Res. Not. IMRN (2007), Art. ID rnm 101.

I know it from the introductory section of

Florian Luca and Juan Luis Varona, *Multiperfect numbers on lines of the Pascal triangle*, Journal of Number Theory, Volume 129, Issue 5 (2009).