MathDB

A positive integer

n>1

is called beautiful if

n

can be written in one and only one way as

n=a_1+a_2+\cdots+a_k=a_1 \cdot a_2 \cdots a_k

for some positive integers

a_1, a_2, \ldots, a_k

, where

k>1

and

a_1 \geq a_2 \geq \cdots \geq a_k

. (For example 6 is beautiful since

6=3 \cdot 2 \cdot 1=3+2+1

, and this is unique. But 8 is not beautiful since

8=4+2+1+1=4 \cdot 2 \cdot 1 \cdot 1

as well as

8=2+2+2+1+1=2 \cdot 2 \cdot 2 \cdot 1 \cdot 1

, so uniqueness is lost.) Find the largest beautiful number less than 100.

Problem Statement