Interesting game. Maybe you know this already, but there is a simple proof that the Nullwertzahlen have density 0: any Nullwertzahl \(n\) satisfies \(n\le2^{2^{\Omega(n)1}}\), so that \(\Omega(n)>\log\log(n)/\log(2)\). As you pointed out in another post, \(\Omega(n)\) is almost always close to \(\log\log{n}\), so very few \(n\) have so many prime factors. Following the proof of the HardyRamanujan theorem, I think you can push this as far as a proof that \(W_0(x)=O(x/(\log{x})^\delta)\) for some \(\delta>0\), but I don't see how to get \(\delta=1\) from just this.
