For integers $n\geq 1$, $H_n$ denotes the $n$th harmonic number. I am looking examples of arithmetic function $a(k)$ and $b(k)$, whose terms are integers $\geq 1$ and such that the terms of this sequence $$a(k)\cdot H_{b(k)},\tag{1}$$ are integers many times, or maybe that such quantity is an infinity.
I know examples of proofs (for example from this site Mathematics Stack Exchange, or proposed in [1] as exercise) showing that the $n$th harmonic number isn't an integer when $n>1$, we know it, for $n\geq 1$, the $$H_n=1+\frac{1}{2}+\ldots+\frac{1}{n}$$ is never an integer when $n>1$. And of course the sequence $$n!\cdot H_n$$ is integer for each $n\geq 1$.
Question. I am trying to find sequences of integers $a(k)$ and $b(k)$ (isn't required number theoretic functions), I am saying a non obvious example of an arithmetic function $a(k)\cdot H_{b(k)}$ satisfying that its general term is integer many times, or maybe even an infinity of such terms are integers. Is it possible? Many thanks.
My no example: An example with few terms being integers is given as $$c(k):=k\cdot H_{\varphi(k)}$$ when $k\geq 1$, with $\varphi(k)$ denoting the Euler's totient function. Thus here $a(k)=k$ and $b(k)=\varphi(k)$. Then I know that the following terms of our sequence $(2)$ are integers: $1, 2, 3, 6$ and $12$. My example seems that has few terms being integers.
References:
[1] Apostol, Introduction to Analytic Number Theory, Springer (1976).
[2] Harmonic Number, from MathWorld.