Eccoci.

Let $a_n$ be a sequence recursively defined by:$\quad a_1=1,\quad\displaystyle a_n=\biggl(\sum_{k=1}^{n-1}k\cdot a_k\biggr)\bmod n$.

So the first $10$ elements of $a_n$ are: $1,1,0,3,0,3,5,4,1,9$.

Let $f(N, M)$ represent the number of pairs $(p, q)$ such that:

$$ \def\htmltext#1{\style{font-family:inherit;}{\text{#1} } } 1\le p\le q\le N \quad\htmltext{and}\quad\biggl(\sum_{i=p}^qa_i\biggr)\bmod M=0 $$

It can be seen that $f(10,10)=4$ with the pairs $(3,3)$, $(5,5)$, $(7,9)$ and $(9,10)$.

You are also given that $f(10^4,10^3)=97158$.

Find $f(10^{12},10^6)$.