Let ${\mathrm{a}}_{\mathrm{n}}={\int }_{-1}^{\mathrm{n}} \left(1+\frac{\mathrm{x}}{2}+\frac{{\mathrm{x}}^2}{2}+\frac{{\mathrm{x}}^3}{3}+\dots \dots +\frac{{\mathrm{x}}^{\mathrm{n}-1}}{\mathrm{n}}\right)\mathrm{dx}$ for $\mathrm{n}\in \mathrm{N}$. Then the sum of all the elements of the set $\left\{\mathrm{n}\in \mathrm{N}:{\mathrm{a}}_{\mathrm{n}}\in (2,30)\right\}$ is ______

$$\begin{gathered} a_n={\int }_{-1}^n\left(1+\frac{x}{2}+\frac{x^2}{3}+...+\frac{x^{n-1}}{n}\right)dx \\ \begin{array}{l}\because a_n={\int }_{-1}^n \left(1+\frac{x}{2}+\frac{x^2}{3}+\dots +\frac{x^{n-1}}{n}\right)dx\\ ={\left[x+\frac{x^2}{2^2}+\frac{x^3}{3^2}+\dots +\frac{x^n}{n^2}\right]}_{-1}^n\\ a_n=\frac{n+1}{1^2}+\frac{n^2-1}{2^2}+\frac{n^3+1}{3^2}+\frac{n^4-1}{4^2}+\dots +\frac{n^n+(-1{)}^{n+1}}{n^2}\\ \text{ Here }{a}_1=2,a_2=\frac{2+1}{1}+\frac{2^2-1}{2}=3+\frac{3}{2}=\frac{9}{2}\\ a_3=4+2+\frac{28}{9}=\frac{100}{9}\\ a_4=5+\frac{15}{4}+\frac{65}{9}+\frac{255}{16}>31\\ \therefore \text{ The required set is }\{2,3\}.\because a_n\in (2,30)\\ \therefore \text{ Sum of elements }=5\end{array} \end{gathered}$$