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Is function p exponential?

Source: Brazilian Mathematical Olympiad 2023, Level U, Problem 1

October 21, 2023

algebrainequalitiesreal analysisfunction

Problem Statement

Let

p

be the potentioral function, from positive integers to positive integers, defined by

p(1) = 1

and

p(n + 1) = p(n)

, if

n + 1

is not a perfect power and

p(n + 1) = (n + 1) \cdot p(n)

, otherwise. Is there a positive integer

N

such that, for all

n > N,

p(n) > 2^n