Here are a few others:
Let $H_n=\sum_{j=1}^n 1/j$. Then for all $n\geq 1$, $$ \sum_{d|n}d\leq H_n+(\log H_n)e^{H_n}. $$ Jeff Lagarias showed that this is equivalent to the Riemann hypothesis!
Let $x_0=2$, $x_{n+1}=x_n-\frac{1}{x_n}$ for $n\geq 0$. Then $x_n$ is unbounded.
The largest integer that cannot be written in the form $xy+xz+yz$, where $x,y,z$ are positive integers, is 462. It is known that there exists at most one such integer $n>462$, which must be greater than $2\cdot 10^{11}$. See J. Borwein and K.-K. S. Choi, On the representations of $xy+yz+xz$, Experiment. Math. 9 (2000), 153-158; https://projecteuclid.org/journals/experimental-mathematics/volume-9/issue-1/On-the-representations-of-xyyzzx/em/1046889597.full.
