Suppose we write $s(n,k)$ for the Stirling numbers of the *first* kind. When $p$ is a prime number, I'm interested in knowing when $s(p^a, k)$ is divisible by $p^a$. So:



> 
> What is known about $s(p^a,k)$ mod $p^a$ ?
> 
> 
> 


I have tried to ask google about this, but the literature is confusing (there are *some* congruences that are proved, but it seems difficult to figure out whether this one is among them). I thought that someone here might know the answer from the top of their head.


(Note that I would really like the answer to be "you need to work it out yourself", for it seems like a fun problem. Unfortunately someone else may have done it already.)


Thanks !


Pierre



congruence for Stirling numbers of the first kind
As said in my comment, this article gives some partial results: Tamás Lengyel, [On p-adic properties of the Stirling numbers of the first kind](http://www.sciencedirect.com/science/article/pii/S0022314X14003102), Journal of Number Theory Volume 148, March 2015, Pages 73–94. 


