\begin{eqnarray*}n^7-n &=& n(n^6-1)\\ &=& n(n^2-1)(n^4+n^2+1)\\ &=& (n-1)n(n+1)(n^4-13n^2+36+14n^2-35)\\ &=& (n-1)n(n+1)[(n^2-4)(n^2-9)+7(2n^2-5)]\\ &=& (n-3)(n-2)(n-1)n(n+1)(n+2)(n+3)+7(2n^2-5)(n^3-n)\end{eqnarray*}