An odd perfect number's third largest prime factor is at most (2N)^(1/6), and its three largest prime factors multiply to at most (2N)^(3/5).
Nielsen, Odd perfect numbers, Diophantine equations, and upper bounds, Mathematics of Computation 84 295 (2015), 2549--2567
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On the third largest prime divisor of an odd perfect number
An odd perfect number's third largest prime factor is at most (2N)^(1/6), and its three largest prime factors multiply to at most (2N)^(3/5).