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.

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}


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