Conservation laws arising from Noether's theorem are varied to obtain a hierarchy of new conservation laws involving perturbations of the fields and their symmetries. The resulting conserved quantities, called Taub numbers, are obstructions to formal power series solutions of the field equations. The main results give conditions for the existence (conservation), for the triviality (vanishing), and for the gauge invariance of these Taub numbers. Gravitational and gauge fields are used to illustrate the theory; examples of other fields to which the theory can be extended are suggested.