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Find the prime factorization of recursive sequence involving maximum

Source: Bundeswettbewerb Mathematik 2020, Round 1 - Problem 4

November 17, 2020

number theoryprime factorizationnumber theory proposed

Problem Statement

Define a sequence

(a_n)

recursively by

a_1=0, a_2=2, a_3=3

and

a_n=\max_{0<d<n} a_d \cdot a_{n-d}

for

n \ge 4

. Determine the prime factorization of

a_{19702020}