A Real Recursion

For every real number $a > 1$ is given the sequence $g_{a}$ by:
$g_{a} (x) = 1$ for $x < a$
$g_{a} (x) = g_{a} (x - 1) + g_{a} (x - a)$ for $x \geq a$
$G (n) = g_{\sqrt{n}} (n)$
$G (90) = 7564511$ .

Find $\sum G (p)$ for $p$ prime and $10000000 < p < 10010000$
Give your answer modulo $1000000007$ .

Write-up coming later

The complete problem is available here. An approach, code, and answer will be added later.