Torsion primes in loop space homology

We construct a finite 1-connected CW complex X such that $H_{\ast }(\mathrm{\Omega }X;Z)$ has p-torsion for the infinitely many primes satisfying p ~ 5,7,17,19 mod 24, but no p-torsion for the infinitely many primes satisfying p ~ 13 or 23 mod 24.