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M. Drmota, C. Krattenthaler:
"A joint central limit theorem for the sum-of-digits function, and asymptotic divisibility of Catalan-like sequences";
Proceedings of the American Mathematical Society, 147 (2019), S. 4123 - 4133.

We prove a central limit theorem for the joint distribution of , , where denotes the sum-of-digits function in base and the 's are positive integers relatively prime to . We do this in fact within the framework of quasi-additive functions. As an application, we show that most elements of ``Catalan-like" sequences--by which we mean integer sequences defined by products/quotients of factorials--are divisible by any given positive integer.

http://dx.doi.org/10.1090/proc/14349