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Source. Printed p. 213, physical PDF p. 2, the paragraph beginning "Denote $\sigma_k(n)$", and its footnote 2, read on the page image.

Statement

With $\sigma_k(n)=\sum_{d\mid n}d^k$, the paper says: "Kac and I conjectured that

$$\sum_{n=1}^{\infty}\frac{\sigma_k(n)}{n!}$$

is irrational for every integer $k>0$. We proved this for $k=1$ and $k=2$, for $k>2$ the proof seems to present great difficulties."

Footnote 2 locates the two proved cases in the problem section of the American Mathematical Monthly: "This was a problem in Amer. Math. Monthly 1 [sic], 264, (1954), for $k=2$ solution by R. Breusch, for $k=1$ solution by J. B. Kelly 60, 557, (1953)." (The volume printed as "1" is the Monthly's 1954 volume, 61.) The paper itself thus attributes the case $k=1$ to Kelly's solution and the case $k=2$ to Breusch's. Neither Monthly item is held in the library, and the paper gives no argument for either case.

Relation to Problem 252

Problem 252 asks exactly this question for every $k\ge1$. The passage is a 1957 statement of the conjecture with the two settled cases named; it is not a proof of anything, and the paper contains no result on the series.

Bears on. #252.