To fill a box of size log(N)/2, put the largest factor into the box. Then put the next largest factor that still fits in the box and insert it. Keep doing this until no more factors fit.
The above will get you the largest factor <= sqrt(N). To get the smallest factor >= sqrt(N), put all the factors in a box, and take them out again largest to smallest, until no more can be taken out. Finally, comapre the two answers to find the closest.
I don't think that does what you think. For example, 2**3 * 19**3 = 54872,
and the square root is 234.277...
Your method gives 19 as the largest less than the square root, and 2**3 * 19**2 = 2888 as the smallest greater than the square root. Notice that 19*2888 = 54872.
A better result is 2**3 * 19 = 152 (because 19**2 = 361).
Quantum Mechanics: The dreams stuff is made of
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