Sequences and Series

Let $< a_{\mathrm{n}} >$ be a sequence such that $a_{1}+a_{2}+\ldots+a_{n}=\frac{n^{2}+3 n}{(n+1)(n+2)}$. If 28 \sum_\limits{k=1}^{10} \frac{1}{a_{k}}=p_{1} p_{2} p_{3} \ldots p_{m}, where $\mathrm{p}_{1}, \mathrm{p}_{2}, \ldots ., \mathrm{p}_{\mathrm{m}}$ are the first $\mathrm{m}$ prime numbers, then $\mathrm{m}$ is equal to