Ap4A phosphorylase catalyzes the phosphorolytic degradation of bis(5'-adenosyl) tetraphosphate (Ap4A) into ADP and ATP. Can also use other Np4N' nucleotides (where N and N' stand for A,C,G or U) as substrates. Cannot catalyze the reverse reaction. Additionally, this enzyme can also catalyze the phosphorolytic degradation of adenosine 5'-phosphosulfate (AMPS) into ADP and sulfate, the reversible exchange reaction between inorganic phosphate and the beta-phosphate of a nucleoside diphosphate (NDP), and the synthesis of Ap4A from AMPS plus ATP. ADP + ATP + H(+) = P(1),P(4)-bis(5'-adenosyl) tetraphosphate + phosphate ADP + H(+) + sulfate = adenosine 5'-phosphosulfate + phosphate Monomer. Belongs to the ATP adenylyltransferase family.