We begin with a succinct proof of Theorem \[infinitelymanystcores\].

Recall that there are infinitely many $g$-cores if $g>1$. For any $g$-core ${\gamma}$, consider the (completely and uniquely defined) partition ${\lambda}$ with ${\mathcal Cor}_g({\lambda})={\gamma}$ and ${\mathcal Quo}_g({\lambda})=({\lambda}_1, \, \ldots , \, {\lambda}_g)=( (1), \, \emptyset, \, \ldots , \, \emptyset)$. Then ${\lambda}_1, \, \ldots , \, {\lambda}_g$ are certainly $(s',t')$-cores (since none of $s'$ and $t'$ is 1), so that, by Theorem \[stcore\], ${\lambda}$ is an $(s'g,t'g)$-core. \[Note that, in order to apply Theorem \[stcore\], we do not actually need the extra hypothesis that $g=\text{gcd}(s'g,t'g)$.\] Also, since ${\mathcal Quo}_g({\lambda}) \neq (  \emptyset, \, \ldots , \, \emptyset)$, the partition ${\lambda}$ is certainly [*not*]{} a $g$-core. Allowing ${\gamma}$ to vary produces an infinite number of (distinct) $(s'g,t'g)$-cores which are not $g$-cores.

Theorem \[yeahtnog\] shows that, if $g \geq 4$ and $t'>1$, then, for any integer $n \geq g$, there exists a $t'g$-core of $n$ that is not a $g$-core. Our next theorem improves this result, particularly if $t' \geq 4$.

\[f\] Fix integers $g\geq 4$ and $t'>1$, and let $t=t'g$. Then, for every $n\geq g$, the number $\hat{\psi}_{t \backslash g}(n)$ of $t \backslash g$-cores of $n$ is bounded below by ${\displaystyle}\sum_{w= 1}^{\left\lfloor 