G-g-generalize

I never meant for it to happen. It violates all intentions of this blog to generalize a theorem. So I’d like to blame Ernie Croot for the suggestion;

Let $B$ be any (unordered) set of integers. 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 $b_i \in B$ for all $i$. Then $\gcd(X_n, L_n) > 1$ necessarily happens infinitely often in exactly $2$ cases: there either exists an odd prime $p$ for which $b_i \equiv b_j \pmod{p}$ for all $b_i, b_j \in B$, or all the $b_i$ are even. And it’s not even necessary to start the sum at $i = 1$.