The 2-adic valuations of Stirling numbers of the second kind

Abstract

In this paper, we investigate the 2-adic valuations of the Stirling numbers $S(n,k)$ of the second kind. We show that $v_2(S(4i,5))=v_2(S(4i+3,5))$ if and only if $i\not\equiv 7(\mathrm{mod}32)$. This confirms a conjecture of Amdeberhan, Manna and Moll raised in 2008. We show also that $v_2(S(2^n+1,k+1))=s_2(n)-1$ for any positive integer $n$, where $s_2(n)$ is the sum of binary digits of $n$. It proves another conjecture of Amdeberhan, Manna and Moll.

arXiv comment9 pages. To appear in International Journal of Number Theory