First-order spectra with one binary predicate

The spectrum, Sp(phi), of a sentence phi is the set of cardinalities of finite structures which satisfy phi. We prove that any set of integers which is in Func(1)(infinity) i.e. in the class of spectra of first-order sentences of type containing only unary function symbols is also in BIN1 i.e, in the class of spectra of first-order sentences of type involving only a single binary relation. We give similar results for generalized spectra and some corollaries: in particular, from the fact that the large complexity class (c) boolean OR NTIME(RAM)(CR) is included in Func(1)(infinity) for unary languages (n denotes the input integer), we deduce that the set of primes and many ''natural'' sets belong to BIN1