We investigate the expressive power of two logics, both with the successor function: first-order logic with an un-interpreted function, and existential monadic second order logic — that is first-order logic over words —, with multiplication by a constant b. We prove that all b-recognizable sets are spectra of those logics. Furthermore, it is proven that some encoding of the set of halting times of a non-deterministic 2-counter automaton is also a spectrum. This yields undecidability of the finite satisfiability problem for those logics. Finally, it is shown that first-order logic with one uninterpreted function and successor can encode quickly increasing functions, such as the Knuth's up-arrows.