Ah! I wasn't sure how sophisticated you were or how much so you wanted to get. Now, I see you know what you want.

Since you don't care what $n is, you may be able to select a set of factors to obtain particular properties. I think that you will have better success going at the problem from an algebraic point of view than searching numerically. These kinds of problems have usually been mined out for many digits.

After Compline,
Zaxo