On a conjecture of Wilf

Let n and k be natural numbers and let S(n,k) denote the Stirling numbers of the second kind. It is a conjecture of Wilf that the alternating sum [...] is nonzero for all n>2. We prove this conjecture for all n≢2 and ≢2944838 mod 3145728 and discuss applications of this result to graph theory, multiplicative partition functions, and the irrationality of p-adic series.