Problem #251: Is\[\sum \frac{p_n}{2^n}\]irrational?

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Progress

Erdős [Er58b] proved that $\sum \frac{p_n^k}{n!}$ is irrational for every $k\geq 1$.

In [Er88c] he further conjectures that $\sum \frac{p_n^k}{2^n}$ is irrational for every $k$, and that if $g_n\geq 2$ and $g_n=o(p_n)$ then\[\sum_{n=1}^\infty \frac{p_n}{g_1\cdots g_n}\]is irrational. (The example $g_n=p_n+1$ shows that some condition on the growth of the $g_n$ is necessary here.)

The decimal expansion of this sum is A098990 on the OEIS.