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Source. Printed p. 94, first paragraph, and the sentence after it. Read on the page image (physical PDF p. 2).

Statement

"Enfin, je tiens à souligner que je n'ai pas réussi à démontrer l'irrationnalité de la somme des séries

$$\sum_{n=1}^{\infty}\frac{1}{p_n\,n!},\qquad \sum_{n=1}^{\infty}\frac{p_n}{2^n}\qquad\text{et}\qquad \sum_{n=1}^{\infty}\frac{1}{p_n\,2^n}\,.$$

Pour ce qui concerne l'irrationnalité des séries semblables, voir [2]."

Reference [2] is P. Erdős, On the irrationality of certain series, Indag. Math. 19 (1957), 212--219, filed as erdos_1957_irrationality_certain_series; that note treats $\sum1/t^{\varphi(n)}$, $\sum1/t^{\sigma(n)}$ and lacunary series $\sum1/t^{n_k}$, not these three.

Relation to problem 251

The second series is exactly problem 251, still open on the site. Section 3 of the paper (p. 99) returns to it as the case $q_n=2$ of the section 3 theorem that "m'échappe entièrement", and the 1988 survey repeats the expectation on its p. 103 ("This is probably very difficult already for $k=1$"). The first and third series are not catalog problems, and this library records no result on either.

Bears on. #251.