Central binomial coefficients

The binomial coefficients can be arranged in triangular form like this:

1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
......

It is known as Pascal's triangle.

The middle coefficients of the even-numbered rows (row numbering starts from zero) form a sequence $a_n(n\ge 0)$: 1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756, .... It has been proven that no number $a_n$ with $n>4$ is squarefree.

Let $f(n)$ be $n$ divided by largest squarefree divisor of $n$. For example, $24=2^3\cdot 3$ and its largest squarefree divisor is $6$, so $f(24)=24/6=4$.

Let $S(N)$ be the sum of all $f(a_n)$ for $n$ from $1$ to $N$. You are given $S(10)=1+1+2+1+6+2+4+3+2+2=24$, $S(100)=429083$ and $S({10}^3)\mathrm{mod}1000000007=258384246$.

Find $S({10}^6)\mathrm{mod}1000000007$.