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Let $n > 0, K > 0$ integers and, for $i \in \{1,...,n\}$, let $k_i$ and $l_i$ be integers such that $k_i + l_i = K$. Assume that for some $i,j \in \{1,...,n\}$ we have $k_i \neq k_j$.

Claim: There exists a sequence $a_1,...,a_n$ of consecutive integers such that $\prod_{i = 1}^n (\frac{a_i}{i})^{k_i} \notin \mathbb{Z}$ or $\prod_{i = 1}^n (\frac{a_i}{i})^{l_i} \notin \mathbb{Z}$. Equivalently, $\sum_{i=1}^n v_p(a_i^{k_i}) < \sum_{i=1}^n v_p(i^{k_i})$ or $\sum_{i=1}^n v_p(a_i^{l_i}) < \sum_{i=1}^n v_p(i^{l_i})$ for some prime $p$, where $v_p(x)$ denotes the $p$-adic valuation of an element $x \in \mathbb{Q}$.

The claim holds true in the cases where $n$ is prime and where $n = p + 1$ for some prime $p$.

Does anybody know a reference dealing with similar problems? Or is this result already known? Are there certain methods to approach such questions?

Thank you very much in advance!

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