**Starting point.** Let $P\subseteq\mathbb{N}$ be the set of primes. By just "looking at" how the first primes are distribute, there is a lot going for the following statement makes a lot of sense:

> For all integers $n,k \geq 2$ we have $$\Big|[2, 2+n]\cap P\Big| \geq \Big|[k, k+n]\cap P\Big|.$$

Interestingly, it is [not known](https://mathoverflow.net/a/259862/8628) whether this statement is true. This inspired the following considerations.

**Taking this to finite arithmetical progressions.** For $a, n\in\mathbb{N}$, let $A^{\leq n}_a=\{j\cdot a: j\in\mathbb{N}, 0\leq j\leq n\}$. If $n\in \mathbb{N}$ and $B\subseteq \mathbb{N}$, let $n+B:= \{n+b: b\in B\}$. Is there a counterexample for the following statement?

> For all integers $a, k, n \geq 2$ **with $a$ odd** we have $$\Big|[2+A^{\leq n}_a\cap P\Big| \geq \Big|[k+A^{\leq n}_a\cap P\Big|.$$