On the sum of digits of the factorial
Abstract
Let b > 1 be an integer and denote by sb(m) the sum of the digits of the positive integer m when is written in base b. We prove that sb(n!) > Cb log n log log log n for each integer n > e, where Cb is a positive constant depending only on b. This improves of a factor log log log n a previous lower bound for sb(n!) given by Luca. We prove also the same inequality but with n! replaced by the least common multiple of 1,2,...,n.
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