This is a question inspired by T. Amdeberhan's recent question, as well as another previos MO question.

For an integer partition $\lambda$, and $k\in \mathbb{N}\cup\{\infty\}$, let $|\lambda|_k$ denote the sum of the parts of $\lambda$, but where we only count each number at most $k$ times. E.g., $|\lambda|_1$ is the sum of the parts of $\lambda$ after removing repeated parts, and $|\lambda|_{\infty}=|\lambda|$ is the usual size of the partition.

Define the coefficients $a_k(n)$ by $$ \frac{\sum_{\lambda} |\lambda|_k \cdot q^{|\lambda|}}{\sum_{\lambda} q^{|\lambda|}} = \left( \sum_{\lambda} |\lambda|_k \cdot q^{\lambda} \right) \cdot \prod_{i=1}^{n} (1-q^i) = \sum_{n\geq 0} a_k(n) q^{n}.$$

From the above-linked questions, we see that $a_1(n) = n$, while $a_{\infty}(n) = \sigma(n) = \sum_{d\mid n} d$, the sum of divisors of $n$. So $a_k(n)$ give a sequence of numbers which "interpolate" between $n$ and $\sigma(n)$ in some sense.

**Question**: What are these $a_k(n)$ for arbitrary $k$? Do they have any nice expression in general?