Problem Archive

Define the sequence $a_1, a_2, a_3, \dots$ as:

• $a_1 = 1$
• $a_{n+1} = 6a_n^2 + 10a_n + 3$ for $n \ge 1$.

Examples:
$a_3 = 2359$
$a_6 = 269221280981320216750489044576319$
$a_6 \bmod 1\,000\,000\,007 = 203064689$
$a_{100} \bmod 1\,000\,000\,007 = 456482974$

Define $B(x,y,n)$ as $\sum (a_n \bmod p)$ for every prime $p$ such that $x \le p \le x+y$.

Examples:
$B(10^9, 10^3, 10^3) = 23674718882$
$B(10^9, 10^3, 10^{15}) = 20731563854$

Find $B(10^9, 10^7, 10^{15})$.