I am studying the compositions $(n\_1,...,n\_r)$ of an integer $m$ such that $i\vert n\_i$ for all $i=1,...,r$. (Recall that a composition $(n\_1,...,n\_r)\vDash m$ of $m$ is just a sequence $(n\_1,...,n\_r)\in \mathbb{P}^r$ of positive integers for which $n\_1+n\_2+\cdots+n\_r=m$.) Here the length $r$ of the composition is not fixed.


Is anything known about these compositions? More specifically, can we relate the number of these compositions to the number of partitions of $m$? Does it have asymptotic expansions that approximate it or recurrence relations that calculate it exactly?


If for whatever reasons it helps, it can be assumed that $m=24$.



Compositions $(n_1,...,n_r)$ of an integer $m$ such that $i$ divides $n_i$
The number $f(m)$ of such sequences is the number $q(m)$ of partitions of $m$ into distinct parts (<https://oeis.org/A000009>). For a bijection, take your sequence $(n\_1,\dots,n\_r)$ and let $\lambda=(\lambda\_1,\dots,\lambda\_r)$ be the conjugate partition (using the standard definition of partition) of the partition with $(n\_i-i)/i$ $i$'s. Then $(\lambda\_1+r,\lambda\_2+r-1,\dots,\lambda\_r+1)$ is a partition of $m$ into distinct parts, and conversely.


An asymptotic formula appears for instance in <https://math.stackexchange.com/questions/2303741>.


