After I've read a problem posed by Édouard Lucas from the first paragraph of the biography from this Wikipedia's article I am trying to get different variations of this problem using number theoretic functions. After I did some experiments (I've calculated some terms using Wolfram Alpha online calculator) I posed a surprising conjecture.
Conjecture. Here $H_n$ denotes the $n$th harmonic number. For each integer $n\geq 1$, the integer $$\sum_{k=1}^n (k!)^2(H_k)^2\tag{1}$$ is a square-free integer (has no repeated primes in its factorization, see this Wikipedia; we get $1$ as $n=1$).
Question 1. Can you refute numerically my conjecture? Can you provide us some reasoning about if our sequence $(1)$ has infinitely many terms being square-free integers? Many thanks.
As you see I tried to get a variation of the problem posed by Lucas, since my proposal in the RHS of my problem is the sequence of square-free integers instead of the square numbers, and the terms of my finite series are the squares of the integers $k!\cdot H_k$. I tried more examples using number theoretic functions.
With the purpose of making this post more interesting I ask to this commmunity about other possible variations, and in order to ask a concise question, I am going to limit the nature of the sequences that are terms in our summation, now the square of sequences of the form $$a(k)\cdot H_k,$$ where $a(k)$ is a sequence of rational numbers. The RHS of your statement/conjecture can be any sequence (I prefer some interesting sequence of figurate numbers, see this MathWorld).
Question 2. Provide us yourself example of a conjecture or statement from a sequence $a(k)$ or rational numbers, such that for each $n>N$ (for a suitable positive integer in your proposal) $N$ $$\sum_{k=1}^n (a(k)\cdot H_k)^2=\text{ a sequence of integers* }.\tag{2}$$ *I prefer figurate numbers, or famous sequences of integers like prime numbers... Many thanks.