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

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

2009-07-20

arxiv.org/abs/0907.3412v2

Mathematics

Number Theory

9 pages. To appear in International Journal of Number Theory

Scientific paper

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 $i\not\equiv 7\pmod {32}$. 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.

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