Best time complexity of sorting numbers in range [1...n log n]
Note in the RAM model, indirection always takes constant time regardless of how large the address is, so each process of radix sort with base $b$ takes $O(n+b)$ time. As a result, the radix sort takes $O((n+b)\log_b(n\log n))=O((n+b)\log_b n)$ time. Choosing $b=n$ makes the asymptotic time linear.

Edit: as suggested by Thinh D. Nguyen, since extracting each digit under base $b$ requires division and modular arithmetic, which are not supported by the standard RAM model, we may want to use the word RAM model instead.