By a theorem of Luck, $K^0 (BS_n) \simeq \mathbb{Z} \times \prod_p (\mathbb{Z}_{p}^\wedge)^{r(p,n)}$ where $r(p,n)$ is the number of partitions of $n$ into powers of $p$ (excluding the trivial partition $1+1 \cdots 1$). Wanting to compute the $r(p,n)$, you can repeat the procedure of making the generating function for all the $n$.

\begin{eqnarray*} \sum (r(p,n)+1)x^n &=& \prod_{j \geq 0} \frac{1}{1 - x^{p^j}}\\ &=& \exp \sum_{j=0}^\infty \sum_{k=1}^{\infty} \frac{x^{kp^j}}{k}\\ \end{eqnarray*}

Last night, I plotted a coarse approximation the quantity inside the exponential for $\mid x\mid <1$. You can start to see the divergences at all the roots of unity.

My question was whether these quantities have some sort of modular properties like $\eta (q)$. Or just estimates of $r(p,n)$ via this method. I am not well versed in this area so I don't know the difficulty of this question or whether this is addressed in the literature.