Hagedorn Temperature of Type I Superstring

In many sources, the Hagedorn temperature of different string theories is given using the high temperature (or high-level limit) $$N\gg1$$ in the calculation of the string entropy such that: $$T_H=\left(\frac{\partial S}{\partial E}\right)^{-1}$$ and the entropy is calculated using the Boltzmann formula for counting partitions $$S(E)=k\log(\Omega(E))$$

For Bosonic string theory for example there are $$B=D-2=24$$ transverse directions, where $$D$$ is the critical dimension, so for level $$N$$ the number of partitions is equivalent to the partition obtained by a Boson with spin $$B$$ in energy level $$E$$: $$\Omega_B(E)=\exp\left(\pi\sqrt{\frac{B\alpha'}{6}}E\right)=_{D=26}\exp\left(4\pi\sqrt{\alpha'}E\right)$$ Where $$\alpha'$$ is the string tension coming from the relation of mass squared (Energy squared) to the level $$N$$.

For Type II superstring theory The number of partitions constitute the partition on each sector on each side (Left and Right moving) as half of states, times degeneracy of $$16$$ coming from the ground state of the $$R$$ sector So we get: $$\Omega_{Type\space II}(E)=64(\Omega_B(E)\Omega_F(E))^2$$ When the bosonic part $$\Omega_B(E)$$ is now taken with spin $$B=D-2=8$$ for the fermionic field the number of partitions is equivalent to the partition obtained by a fermion with spin $$F$$ in energy level $$E$$: $$\Omega_F(E)=\exp\left(\pi\sqrt{\frac{F\alpha'}{12}}E\right)=_{D=10}\exp\left(\pi\sqrt{\frac{2\alpha'}{3}}E\right)$$ So overall we have $$\Omega_{Type\space II}(E)\approx\exp\left(2\pi\sqrt{\frac{(F+2B)\alpha'}{12}}E\right)=_{D=10}\exp\left(2\sqrt{2}\pi\sqrt{\alpha'}E\right)$$

For Heterotic superstring theory The left moving sector is given by a Bosonic string in $$\tilde{D}=26$$ dimensions while the right moving sector is given by a supersymmetric string in $$D=10$$ dimensions, taking this into account the number of partitions would be: $$\Omega_{Heterotic}(E)=16(\Omega_\tilde{B}(E))^{\tilde{D}=26}(\Omega_B(E)\Omega_F(E))^{D=10}$$ That is: $$\Omega_{Heterotic}(E)\approx\exp\left(\pi\sqrt{\frac{(F+2B+2\tilde{B})\alpha'}{12}}E\right)=_{\tilde{D}=26,D=10}\exp\left(2\pi\left(1+\frac{1}{\sqrt{2}}\right)\sqrt{\alpha'}E\right)$$

Unfortunately, I didn't find in any source explicitly how to derive the number of the partition in order to estimate the entropy thus Hagedorn temperature of Type I superstring theory. According to Wikipedia: "Type I string theory can be obtained as an orientifold of type IIB string theory, with 32 half-D9-branes added in the vacuum to cancel various anomalies giving it a gauge group of SO(32) via Chan-Paton factors". Is this means in particular that Type I superstrings have the same number of partitions so the same entropy and Hagedorn temperature as Type II superstrings?