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Let Fn be the nth Fibonacci number. The order of appearancez(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

On Integer Numbers with Locally Smallest Order of Appearance in the Fibonacci Sequence