A number is called special if it has mutually distinct exponents in its canonical prime factorizaton for all exponents. In this paper, we prove Bombieri–Vinogradov and Barban–Davenport–Halberstam type theorems for special numbers. As applications, we get the asymptotic formulae for the Tichmarsh divisor problem for the special numbers and a binary quadratic Titchmarsh divisor problem for the special numbers.