Let $[n]:=\{1,\cdots,n\}$. It is known that $\{\log(p) | p \text{ is prime }\}$$\{\log(p) \mid p \text{ is prime }\}$ is linearly independent over $\mathbb{Q}$. For a subset $A \subset [n]$ we can consider the matrix $L(A):=(\log(x) | x \in A)$$L(A):=(\log(x) \mid x \in A)$ with $\text{ rank}_{\mathbb{Q}}(L(A)) = \omega(\prod_{x \in A} x )$$\operatorname{rank}_{\mathbb{Q}}(L(A)) = \omega(\prod_{x \in A} x )$, where $\omega(n)$ counts the distinct prime divisors of $n$. Then we have:
$$\sum_{ A \subset [n]} (-1)^{|A|} \omega(\prod_{x \in A} x) = 0$$$$\sum_{ A \subset [n]} (-1)^{|A|} \omega\left(\prod_{x \in A} x\right) = 0$$
Which denoting $(n;k) := \#\{A \subset [n]| \omega(\prod_{x \in A} x) = k\}$$(n;k) := \#\{A \subset [n]\mid \omega(\prod_{x \in A} x) = k\}$ for $k=0,\cdots,n$ might be rewritten as:
$$\sum_{k=0}^n (-1)^k (n;k)=0$$
My question is, if this last quantity can be interpreted as one Euler characteristic [https://en.wikipedia.org/wiki/Euler_characteristic#Topological_definition]?
Here are some numbers $(n;k)$ computed:
1 [2]
2 [2, 2]
3 [2, 4, 2]
4 [2, 8, 6, 0]
5 [2, 10, 14, 6, 0]
6 [2, 10, 30, 22, 0, 0]
7 [2, 12, 40, 52, 22, 0, 0]
8 [2, 20, 80, 108, 46, 0, 0, 0]
9 [2, 24, 148, 232, 106, 0, 0, 0, 0]
10 [2, 24, 180, 488, 330, 0, 0, 0, 0, 0]
11 [2, 26, 204, 668, 818, 330, 0, 0, 0, 0, 0]
12 [2, 26, 332, 1308, 1714, 714, 0, 0, 0, 0, 0, 0]
13 [2, 28, 358, 1640, 3022, 2428, 714, 0, 0, 0, 0, 0, 0]
14 [2, 28, 390, 2280, 5646, 5884, 2154, 0, 0, 0, 0, 0, 0, 0]
Thanks for your help.
