For the given element, like oxygen, the number of neutrons may vary. But no matter what the number of neutrons, the element keeps the identity of it, based on the atomic number. Differing numbers of neutrons result in a number of isotopes for the given element.
Each element has 1 particular isotope which is found in nature often. But all the elements have several isotopes. Changing the number of neutrons in an element's nucleus results in a difference in weight, and a difference in the density also, of the element. Thus, hydrogen containing a neutron or two in the nucleus, along with the proton, is called as heavy hydrogen.
Common carbon has an atomic weight of around 12, and is called as carbon 12 or C12. But sometimes it has an atomic weight of about 14, and is called as carbon 14 or C14.