Lucky numbers properties

Ulam

Let be [tex]l_n[/tex] the n-th lucky number (Ulam sieve) and
[tex]J_n=\{k \in \mathbb{N} :k \le n \}[/tex]. So, is it possible to have a proof that exists a

[tex]u_n(k): J_n \longrightarrow \{-1,+1\}[/tex] such that

[tex]n=\sum_{k=1}^{n}u_n(k)l_k[/tex]

for all [tex]n[/tex] ?