## Another generalization

Let $L_n$ be the least common multiple of $\{1,2,..,n\}$ and $X_n = L_n \displaystyle \sum_{i=1}^n \dfrac{b_i}{i}$, where $b_{i+m}$ $= b_i \in \aleph$ for all $i$ and some $m$. Then $\gcd(X_n, L_n) > 1$ necessarily happens infinitely often.

Proof.
Assume a $p > a$ exists that divides $X_a = L_a \displaystyle \sum_{i=1}^a \dfrac{b_i}{i}$ for some $a \ge m$. Note that we may assume so, because if such a $p$ doesn’t exist, we’re immediately done since every primedivisor of $X_a$ must then be smaller than or equal to $a$ and thus also be a divisor of $L_a$. Let $k$ be any multiple of $\varphi(m)$ and $n = ap^k$. We then have:

$X_n = L_n \displaystyle \sum_{i=1}^n \dfrac{b_i}{i} \equiv L_n \displaystyle \sum_{i=1}^a \dfrac{b_{ip^k}}{ip^k} \equiv \dfrac{L_n}{p^k} \displaystyle \sum_{i=1}^a \dfrac{b_i}{i} \equiv 0 \pmod{p}$