2023 IOQM/Problem 29

Problem

A positive integer $n > 1$ is called $beautiful$ if $n$ can be written in one and only one way as $n = a_1 + a_2 +... + a_k = a_1 a_2 ... a_k$ for some positive integers $a_1, a_2, . . . , a_k$ , where $k > 1$ and $a_1 \geq a_2 \geq ... \geq a_k$ . (For example 6 is beautiful since 6 = 3 · 2 · 1 = 3 + 2 + 1 , and this is unique. But 8 is not beautiful since 8 = 4 + 2 + 1 + 1 = 4 · 2 · 1 · 1 as well as 8 = 2 + 2 + 2 + 1 + 1 = 2 · 2 · 2 · 1 · 1 , so uniqueness is lost.) Find the largest beautiful number less than 100.

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2023 IOQM/Problem 29

Problem