Alspach got 10,019 using a math package that computed Stirling numbers of the second kind. The Stirling numbers are the number of ways to partition a set of n elements into k disjoint subsets. Let me know if you figure out how this works, I'm still looking into it. I suspect you can use inclusion-exclusion for each value of n, where each term represents the number of ways to get at least k repeats. This would explain why you need software.

Are you sure Sklansky got this same number? If you don't carry all the significant digits in the approximation method your answer can be off considerably.