A104588 Product of primes less than or equal to sqrt(n).
1, 1, 1, 2, 2, 2, 2, 2, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 210, 210, 210, 210, 210, 210, 210, 210, 210, 210, 210, 210, 210, 210, 210, 210, 210, 210, 210, 210, 210, 210
http://oeis.org/A104588
seems to be larger than $n$ for $n > 48$, so there is a very finite number of cases to test.
That is not a complete proof but it does explain one reason the result should be a short list of small $n$.
Another sledgehammer is http://oeis.org/A003418
A003418 a(n) = least common multiple (lcm) of $\{1, 2, ..., n\}$
1, 1, 2, 6, 12, 60, 60, 420, 840, 2520, 2520, 27720, 27720, 360360, 360360, 360360, 720720, 12252240, 12252240, 232792560, 232792560, 232792560, 232792560, 5354228880, 5354228880, 26771144400, 26771144400, 80313433200, 80313433200
which becomes much larger than $n^2$ according to the remark
An assertion equivalent to the Riemann hypothesis is: $| \log(a(n)) - n | < \sqrt{n} \log(n)^2$ (for $n \geq 3$).