  While observations of large-scale structure and the cosmic microwave
background (CMB) provide strong constraints on the amplitude of the primordial
power spectrum (PPS) on scales larger than 10~Mpc, the amplitude of the power
spectrum on sub-galactic length scales is much more poorly constrained. We
study early structure formation in a cosmological model with a blue-tilted PPS.
We assume that the standard scale-invariant PPS is modified at small length
scales as $P(k) \sim k^{m_{\rm s}}$ with $m_{\rm s} > 1$. We run a series of
cosmological hydrodynamic simulations to examine the dependence of the
formation epoch and the characteristic mass of primordial stars on the tilt of
the PPS. In models with $m_{\rm s} > 1$, star-forming gas clouds are formed at
$z > 100$, when formation of hydrogen molecules is inefficient because the
intense CMB radiation destroys chemical intermediates. Without efficient
coolant, the gas clouds gravitationally contract while keeping a high
temperature. The protostars formed in such "hot" clouds grow very rapidly by
accretion to become extremely massive stars that may leave massive black holes
with a few hundred solar-masses at $z > 100$. The shape of the PPS critically
affects the properties and the formation epoch of the first generation of
stars. Future experiments of the CMB polarization and the spectrum distortion
may provide important information on the nature of the first stars and their
formation epoch, and hence on the shape of the small-scale power spectrum.
