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Source. C. Badea, The irrationality of certain infinite series, Glasgow Math. J. 29 (1987), no. 2, 221--228, doi:10.1017/S0017089500006868. Corollary 1 and the remark after it are on p. 224 (Section 3). Bibliographic details are on the source card.

Statement

Corollary 1 (p. 224). Let $(a_n)$, $n\ge1$, be a sequence of positive integers such that

$$a_{n+1}>a_n^2-a_n+1$$

(the paper's (8)) holds for all large $n$. Then $\sum_{n\ge1}1/a_n$ is irrational.

The paper relates it to the theorem of Erdős and Straus (its Theorem A, p. 221), which assumes $a_{n+1}\ge a_1a_2\cdots a_n$ for every $n$ and $a_{n+1}\ne a_n^2-a_n+1$ for infinitely many $n$.

Sharpness (remark, p. 224). For $c_1=2$ and $c_{n+1}=c_n^2-c_n+1$ the paper records $\sum_{n\le k}1/c_n=1-(c_{k+1}-1)^{-1}$, hence $\sum_{n\ge1}1/c_n=1$, so (8) cannot be replaced by $a_{n+1}\ge a_n^2-a_n+1$. The paper calls Corollary 1 best possible in a certain sense and notes that this example answers the last question of Problem E.24 in Guy's Unsolved problems in number theory (1981) negatively.

Proof pointer

P. 224: take $b_n=1$ in the Theorem.

Read depth. Claims checked: the statement and the sharpness remark were read clause by clause on p. 224 of the print.

Dependencies

Theorem (p. 222).

Bears on

• Problem 243: by contraposition, a sequence of positive integers with rational reciprocal sum has $a_{n+1}\le a_n^2-a_n+1$ for infinitely many $n$. The problem asks for equality for all large $n$ under its hypotheses, which the corollary does not give. The sharpness example is the problem's recurrence.
• Problem 267: the paper proves Corollary 4 by checking (8) for $a_n=F_{2^n+1}$, which gives the instance $n_k=2^k+1$ of the problem.