  We study primordial black holes (PBHs) formation in the excursion set theory
(EST) in a vast range of PBHs masses with and without confirmed constraints on
their abundance. In this work, a new concept of the first touch in the EST is
introduced for PBHs formation which takes into account the earlier horizon
reentry of smaller masses. Our study shows that in the EST, it is possible to
produce PBHs in different mass ranges which could make up all dark matter. We
also show that in a broad blue-tilted power spectrum, the production of PBHs is
dominated towards a smaller mass. Our analysis put upper limit $\sim\,$0.1 on
the amplitude of the curvature power spectrum at length scales relevant for
PBHs formation.
