# Problem 593: Fleeting Medians
We define two sequences \$S = \\{S(1), S(2), ..., S(n)\\}\$ and \$S\_2 =
\\{S\_2(1), S\_2(2), ..., S\_2(n)\\}\$: \$S(k) = (p\_k)\^k\$ mod
\$10007\$ where \$p\_k\$ is the \$k\$th prime number. \$S\_2(k) = S(k) +
S(\\lfloor\\frac{k}{10000}\\rfloor + 1)\$ where \$\\lfloor \\cdot
\\rfloor\$ denotes the floor function. Then let \$M(i, j)\$ be the
median of elements \$S\_2(i)\$ through \$S\_2(j)\$, inclusive. For
example, \$M(1, 10) = 2021.5\$ and \$M(10\^2, 10\^3) = 4715.0\$. Let
\$F(n, k) = \\sum\_{i=1}\^{n-k+1} M(i, i + k - 1)\$. For example,
\$F(100, 10) = 463628.5\$ and \$F(10\^5, 10\^4) = 675348207.5\$. Find
\$F(10\^7, 10\^5)\$. If the sum is not an integer, use \$.5\$ to denote
a half. Otherwise, use \$.0\$ instead.