Let $S_m=(x_1,x_2,\dots ,x_m)$ be the $m$-tuple of positive real numbers with $x_1+x_2+\cdots +x_m=m$ for which $P_m=x_1\cdot x_2^2\cdot \cdots \cdot x_m^m$ is maximised.

For example, it can be verified that $\lfloor P_{10}\rfloor =4112$ ($\lfloor \rfloor$ is the integer part function).

Find $\sum _{m=2}^{15}\lfloor P_m\rfloor$.