6 problems

Let $a(n)$ and $b(n)$ be the previously defined sums associated with the even and odd numbers, respectively. Bounds conjecture. For every $n\in\mathbb{N}$, … and for every $n\geq5$…

Upper-bound conjecture. For every $n\in\mathbb{N}\setminus\{10^k-1:k\in\mathbb{Z}^{+}\}$,

Fix an integer base $q>1$. A product of $k$ digits greater than $1$ for base $q$ means a product of $k$ elements of $\{2,\ldots,q-1\}$. Zero-digit conjecture for products in base…

Let $S_{10}$ be the Sloane map, sending a nonnegative integer to the product of its base-$10$ digits, and let $u{10}(n)$ be the least number of iterations needed for $S{10}^m(n)$…

Powertrain fixed-point conjecture. These are the only numbers fixed under the powertrain map.

Persistence finiteness conjecture. The sequence of smallest numbers having persistence $1,2,\ldots$ is finite; equivalently, no number has persistence greater than $11$.