Let a ≥ 2 be an integer and n be the greatest integer such that a is a n-th power of an integer. We prove that the Hausdorff dimension of the set of numbers which are simply normal to base a, not simply normal to any base am/n with m > n, not simply normal to any base am/n with m < n, m does not divide n and normal to each algebraic base multiplicatively independent to base a, is equal to 1. This extends a previous result of P. Hertling [9] who proved that a set bigger than the above set has the cardinality of the continuum. For the proof we use the results of [2] and [3] and some probability measures constructed by an inhomogeneous Markov process.