I don't know this result about triangular numbers. But first of all, I don't agree with your value of .

Perhaps you've already learned that or, more precisely, where is called the Euler constant. Due to that, the least possible number you are looking for is probably near .

In fact, I checked with Maple that the least is 12367, so this estimate is pretty precise...

There is a nice (classical) method on this wikipedia page that gives you a loose choice for : , hence is OK. But not very close to the least possible choice...