The favourite numbers of Santa Claus

Saint Nicholas enjoyed playing with numbers and digits. His favourite was the digit $1$, because the New Year begins on $1.01$.

Years passed, but he remained superstitious—he did not like numbers where $3$ follows $1$, that is, the number $13$. For the New Year, he decided to give a new task: count how many of Saint Nicholas's favourite prime numbers are contained in the interval $[a,b]$.