On integer numbers with locally smallest order of appearance in the Fibonacci sequence

Let Fn be the nth Fibonacci number. The order of appearance z n of a natural number n is defined as the smallest natural number k such that n divides Fk. For instance, for all n Fm ≥ 5, we have z n − 1 >z n <z n 1 . In this paper, we will construct infinitely many natural numbers satisfying the previous inequalities and which do not belong to the Fibonacci sequence