Existence of nontrivial μ-Sondow numbers above |μ|

Let

$$mu be an integer, and let$$

\mathfrak{S}_{$mu} denote the set of

$$mu-Sondow numbers. **Existence conjecture.** For every$$

mu $in$\mathbb{Z}$,

(|$mu|,\infty)\cap\mathfrak{S}_{$mu}\neq\emptyset.

The paper explains that 1\in\mathfrak{S}_{mu}$ for every

mu$, but the existence of any

mu-Sondow number other than $1$ is open; this conjecture asserts such an example can always be found greater than $|$mu|.