The atomic number of an atom corresponds to the number of protons it contains. Protons are positively charged particles found in the nucleus of an atom. Neutrons, on the other hand, have no charge and contribute to the mass of the atom. Since you mentioned that the atom has no protons or neutrons, it means it has no nucleus at all.
In such a case, the atom would not have any atomic number. The atomic number is a fundamental property of elements, and without any protons, the atom would not be associated with any specific element. Atoms with no protons or neutrons are extremely rare and unstable, as the strong force that holds the nucleus together is absent.