which kind of radioactive decay can change the atomic number of a
nucleus without changing its mass number?

Beta decay, which converts between protons and neutrons.  (This is an
oversimplification -- there's typically also an electron or positron
and a neutrino involved.)  It has the effect of changing the atomic
number (number of protons), while leaving the mass number (protons
plus neutrons) unchanged.

Here's a wikipedia article on the topic: http://en.wikipedia.org/wiki/Beta_decay

Beta decay. 

In beta decay one of the neutrons in the nucleus loses a high-energy
electron, transforming into a proton. Thus the atomic number gains one
while the mass number stays the same.