  We formulate conjectural blowup formulas for Segre and Verlinde numbers on
moduli spaces of sheaves on projective surfaces $S$ with $p_g(S)>0$ and
$b_1(S)=0$. As applications we give a give a conjectural formula for the
Donaldson invariants of $S$ in arbitrary rank, as well as for the $K$-theoretic
Donaldson invariants, and some Donaldson invariants with fundamental matters.
