I think the method I sketched does not do what you think :-)
Lets take your example. The square root is 234.277. The largest prime factor is 19. The largest remaining factor is 19, but 19*19 > 234.77, so we try the next largest factor, which is 2. 19*2 <= 234.77, so we keep it. The next largest is 2, and 19*2*2 <= 234.77, so we keep it. The next largest (and final) factor is 2, and 19*2*2*2 = 152 < 234.77, so we stop there. 152 is our best attempt for factor closest to but <= sqrt(54872).
A little thought shows that the greedy method from above will give 54872/152 = 361. If one uses a linear distance, then 152 is the closest factor by the grredy heuristic.
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