**Foias constant** also comes from a mistake. I quote directly from *An interesting serendipitous real number* by John Ewing and Ciprias Foias:

This is the story of a remarkable real number, the discovery of which was
due to a misprint. Namely, in the midseventies, while Ciprian was at the
University of Bucharest, one of his former students approached him with
the following question:

(Q) If $x_1>0$ and $x_{n+1}=\left(1+\frac1{x_n}\right)^n$ $(n= 1,2, ... )$, can $x_n\rightarrow\infty$?

This was listed in a fall issue of the "Gazeta Matematica" as one of
the problems given at the previous summer admission examination for
prospective freshmen in the Department of Mathematics at the University
of Bucharest. Ciprian found the answer in about one day, but considered
that the problem was even above the sophomore level. He also found that
(Q) is a misprinted version of the following question (given by Professor N.
Boboc):

(Q') If $x_1>0$ and $x_{n+1}=\left(1+\frac1{x_n}\right)^{x_n}$ $(n= 1,2, ... )$, can $x_n\rightarrow\infty$?

This was an appropriate exam question since the answer is clearly "No."
Years later, in the 1980s, Ciprian told the story to Professor P. Halmos,
who in turn told the story to John, but mischieveously did not mention
at all Ciprian's answer to (Q), so a day later John also found the answer.
This answer is given by the following

**Theorem 1.1** *There exists exactly one real number $a\sim 1.187$ such that
if $x_1=a$ then $x_n\rightarrow\infty$. Moreover in this case
$$x_n\frac{\ln n}n\rightarrow 1 \text{ for } n\rightarrow\infty \ \ (1)$$*
Relation (1) can be rewritten as
$$\lim_{n\rightarrow\infty} \frac{x_n}{\pi(n)}=1 \ \ (2)$$
where $\pi(n)$ is the number of primes less than $n$. However after many attempts to establish a deeper connection with the Prime Number Theorem, we came to believe that relation (2) is fortuitous. A strong argument for this opinion is provided by Theorem 3.1 below in which we show that the estimate for the error
$$x_n\frac{\ln n}n-1$$
differs from its analog in the Prime Number Theorem.

*An interesting serendipitous real number*. J. Ewing, C. Foias.
*Finite versus infinite. Contributions to an eternal dilemma*. Springer (2000), pages 119-126.