We prove a formula which relates Euler characteristic of moduli spaces of stable pairs on local K3 surfaces to counting invariants
of semistable sheaves on them. Our formula generalizes KawaiYoshioka’s formula for stable pairs with irreducible curve classes
to arbitrary curve classes. We also propose a conjectural multiple cover formula of sheaf counting invariants which, combined with
our main result, leads to an Euler characteristic version of KatzKlemm-Vafa conjecture for stable pairs.