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Dec 4, 2013 at 17:14 comment added The Masked Avenger Now consider the power of a prime dividing a central binomial coefficient. You should get an almost guaranteed upper bound of N/2 plus c logN for c less than 1. In fact plus c loglogN seems likely.
Dec 4, 2013 at 6:39 history answered mathlove CC BY-SA 3.0