The sum of the $k$^th powers of the first $n$ positive integers can be expressed as a polynomial of degree $k+1$ with rational coefficients, the Faulhaber's Formulas:
$1^k+2^k+...+n^k=\sum _{i=1}^n{i}^k=\sum _{i=1}^{k+1}{a}_i{n}^i=a_{1}n+a_2{n}^2+...+a_k{n}^k+a_{k+1}{n}^{k+1}$,
where $a_i$'s are rational coefficients that can be written as reduced fractions $p_i/q_i$ (if $a_i=0$, we shall consider $q_i=1$).

For example, $1^4+2^4+...+n^4=-\frac{1}{30}n+\frac{1}{3}{n}^3+\frac{1}{2}{n}^4+\frac{1}{5}{n}^5.$

Define $D(k)$ as the value of $q_1$ for the sum of $k$^th powers (i.e. the denominator of the reduced fraction $a_1$).
Define $F(m)$ as the $m$^th value of $k\ge 1$ for which $D(k)=20010$.
You are given $D(4)=30$ (since $a_1=-1/30$), $D(308)=20010$, $F(1)=308$, $F(10)=96404$.

Find $F({10}^5)$.