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How can I find a function (or more) which satisfy $f(n)=f(f(n-1))=f^n(1)$, defined for positive integers n, such that it is a bijection between the positive integers? And which satisfy i) For every integer k, f(n)=n+k has only finitely many solutions in n.

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If $f$ is bijective, it follows immediately from $f(n)=f(f(n-1))$ that $n=f(n-1)$. This doesn't establish a bijection since there's nothing left to be mapped to $1$; thus there is no such $f$, and you don't need the second part of the equation to show this.