  We propose a systematic approach to computing the BPS spectra of any 5d/6d
supersymmetric quantum field theory in Coulomb phases, which admits either
gauge theory descriptions or geometric descriptions, based on the
Nakajima-Yoshioka's blowup equations. We provide a significant generalization
of the blowup equation approach in terms of both properly quantized magnetic
fluxes on the blowup $\hat{\mathbb{C}}^2$ and the effective prepotential for
5d/6d field theories on the Omega background which is uniquely determined by
the Chern-Simons couplings on their Coulomb branches. We employ our method to
compute BPS spectra of all rank-1 and rank-2 5d Kaluza-Klein (KK) theories
descending from 6d $\mathcal{N}=(1,0)$ superconformal field theories (SCFTs)
compactified on a circle with/without twist. We also discuss various 5d SCFTs
and KK theories of higher ranks, which include a few exotic cases such as new
rank-1 and rank-2 5d SCFTs engineered with frozen singularity as well as the 5d
$SU(3)_8$ gauge theory currently having neither a brane web nor a smooth
shrinkable geometric description. The results serve as non-trivial checks for a
large class of non-trivial dualities among 5d theories and also as independent
evidences for the existence of certain exotic theories.
