For each positive integer $n$, let $p(n)$ be the number of partitions of $n$ (i.e., the number of ways to write $n$ as a sum of positive integers), and let $q(n)$ be the number of strict partitions of $n$ (i.e., the number of ways to write $n$ as a sum of distinct positive integers).The integer sequences $p(n)\ (n=1,2,3,\ldots)$ and $q(n)\ (n=1,2,3,\ldots)$ are
available from http://oeis.org/A000041 and http://oeis.org/A000009, respectively. The famous Hardy-Ramanujan formula states that
$$p(n)\sim\frac{e^{\pi\sqrt{2n/3}}}{4\sqrt3}\qquad \ \text{as}\ \ n\to+\infty.$$
It is also known that $$q(n)\sim\frac{e^{\pi\sqrt{n/3}}}{4(3n^3)^{1/4}}\qquad \ \text{as}\ \ n\to+\infty.$$
Thus the two series
$$\sum_{n=1}^\infty\frac1{p(n)}\quad\mbox{and}\quad \sum_{n=1}^\infty\frac1{q(n)}$$
converge.
Question. Are the numbers $$s=\sum_{n=1}^\infty\frac1{p(n)}\quad\mbox{and}\quad t=\sum_{n=1}^\infty\frac1{q(n)}$$ transcendental?
I conjecture that both $s$ and $t$ are transcendental numbers. Any way to prove this? If not, can one prove that $s$ and $t$ are irrational numbers?
Your comments are welcome!
