MathDB

Let

\mathbb{N}

be the set of positive integers. Determine all positive integers

k

for which there exist functions

f:\mathbb{N} \to \mathbb{N}

and

g: \mathbb{N}\to \mathbb{N}

such that

g

assumes infinitely many values and such that

f^{g(n)}(n)=f(n)+k

holds for every positive integer

n

.
(Remark. Here,

f^{i}

denotes the function

f

applied

i

times i.e

f^{i}(j)=f(f(\dots f(j)\dots ))

Problem Statement