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Let $\mathbb{N} = \{0,1,2,\dots\}$.
For $k \in \mathbb N$,
let $B_{2k} \in \mathbb{Q}$ denote the $(2k)$\textsuperscript{th} Bernoulli number.
For $m \in \mathbb{N}$, we say that a prime $p$ is $m$-regular if
$p$ is odd and $p$ doesn't divide the numerator of $B_{2k}$
for each $k \in \mathbb N$ satisfying
$1 \le 2k \le \min\left( m, p-3\right)$.

Fix a real number $\alpha > 1/2$ and define
\[ M_{\alpha}(p) := \left\lfloor \frac{\sqrt{p}}{(\log p)^{\alpha}}\right\rfloor \in \mathbb{N}. \]
Prove that for real numbers $X \to +\infty$,
\[
  \#\left\{ p\le X \text{ prime}: p \text{ is not $M_{\alpha}(p)$-regular} \right\}
  = O\left( \frac{X}{(\log X)^{2 \alpha}} \right)
\]
where the implied constant may depend on $\alpha$.

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