Lemma: Let $n,a$, be positive integers with $r = n \bmod a$, the remainder after dividing $n$ by $a$. Using Iverson notation [statement] is $1$ if true, $0$ otherwise, $\lfloor 2n/a \rfloor = 2\lfloor n/a \rfloor + [2r \geq a]$.

From this, one shows that the exponent of a prime $p$ in a prime factorization of $C_n={n \choose {\lfloor n/2 \rfloor}}$ is at most $\log_p n$, and is more likely to be about half of that if it is nonzero, by looking at the base $p$ expansion of $n$.

Now suppose $n = c + \lfloor N/2 \rfloor$ and $a_n \gt 1$. Then $P= n!/(n-c)!$ is such that $P^2$ divides $C_N$. For large $N$, $c$ should be bounded by $\min_{p^2 \leq N} pe_p/2$ where $e_p$ is the largest power of $p$ dividing $C_n$. An upper bound that should be tight in general is to look at $p=2,3$ and $5$ and use the minimum based on those. For other bounds, find a prime $q$ with $q^2 \gt N$ and $N/2 \bmod q$ close to $q$: the next multiple of $q$ after $N/2$ is also an upper bound. I do not know how to find $\min(N)$ algebraically, but it is clear one does not have to look far from $N/2$.

There are similar problems depending on sequences of smooth numbers as well as divisibility by small primes. In addition to one-complexity of an integer, my MathOverflow favorite can be found at Factorials in Pascal's triangle. Perhaps a compilation of these can be solved with an appropriate literature search.

Gerhard "Ask Me About Small Factors" Paseman, 2013.12.04

Lemma: Let $n,a$, be positive integers with $r = n \bmod a$, the remainder after dividing $n$ by $a$. Using Iverson notation [statement] is $1$ if true, $0$ otherwise, $\lfloor 2n/a \rfloor = 2\lfloor n/a \rfloor + [2r \geq a]$.

From this, one shows that the exponent of a prime $p$ in a prime factorization of $C_n={n \choose {\lfloor n/2 \rfloor}}$ is at most $\log_p n$, and is more likely to be about half of that if it is nonzero, by looking at the base $p$ expansion of $n$.

Now suppose $n = c + \lfloor N/2 \rfloor$ and $a_n \gt 1$. Then $P= n!/(n-c)!$ is such that $P^2$ divides $C_N$. For large $N$, $c$ should be bounded by $\min_{p^2 \leq N} pe_p/2$ where $e_p$ is the largest power of $p$ dividing $C_n$. An upper bound that should be tight in general is to look at $p=2,3$ and $5$ and use the minimum based on those. For other bounds, find a prime $q$ with $q^2 \gt N$ and $N/2 \bmod q$ close to $q$: the next multiple of $q$ after $N/2$ is also an upper bound. I do not know how to find $\min(N)$ algebraically, but it is clear one does not have to look far from $N/2$.

There are similar problems depending on sequences of smooth numbers as well as divisibility by small primes. In addition to one-complexity of an integer, my MathOverflow favorite can be found at Factorials in Pascal's triangle. Perhaps a compilation of these can be solved with an appropriate literature search.

Gerhard "Ask Me About Small Factors" Paseman, 2013.12.04