Let $\kappa:\lbrace 1,2,3,\ldots\rbrace\longrightarrow \mathbb Z$ be the convolutional inverse in the Dirichlet ring of $n\longmapsto {n+1\choose 2}$. It is defined by $\kappa(1)=1$ and by the functional equation $\sum_{d\vert n}{d+1\choose 2}\kappa(n/d)=0$ for $n\geq 2$.

It is not hard to show that $\kappa(3\cdot 5\cdot p^2)=0$ for every prime number $p\geq 7$.

There seem to be very few other values $n$ such that $\kappa(n)=0$. My computer found only two : $2^4\cdot 13\cdot 23$ and $5^3\cdot 7\cdot 37$.

*Is the set of such exceptional zeros of $\kappa$ finite or not?*