For each integer $n\geq 1$ we consider the arithmetic function $$S(n)=\sum_{k=1}^n n\text{ mod }k,\tag{1}$$
the sum of remainders function, the arithmetic function *A004125* from the OEIS.

**Example.** We've that for $n=6$ $$S(6)=0+0+0+6\text{ mod }4+6\text{ mod }5+0=2+1=3.$$

This arithmetic function was studied for example in [1]. I wondered about a type of problems that are in the literature, that is in our case what about the irrationality of the real number $$\sum_{n=1}^\infty\frac{S(n)}{n!}.\tag{2}$$

I don't know if this previous example is in the literature or has good mathematical content.

Question.Is it possible to deduce that $$\sum_{n=1}^\infty\frac{S(n)}{n!}$$ is irrational? Or well, is it possible to discard it as an irrational?

I am asking about if it possible to do or provide some work, reasonings or heuristics, about it. Then I should to accept an answer. If it is in the literature refer the article and I try to search and read the statement and proof.

## References:

[1] Michael Z. Spivey, *The Humble Sum of Remainders Function*, Mathematics Magazine, Vol. **78**, No. 4 (Oct., 2005).

On the irrationality of a divisor function series, INTEGERS: Electronic Journal of Combinatorial Number Theory7, article #A31, (2007). $\endgroup$ – user142929 Sep 1 '19 at 10:43