Smallest Triplet of Integers whose Product with Divisor Count are Equal

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Theorem

Let $\map {\sigma_0} n$ denote the divisor count function: the number of divisors of $n$.

The smallest set of $3$ integers $T$ such that $m \, \map {\sigma_0} m$ is equal for each $m \in T$ is:

$\set {168, 192, 224}$


Proof

\(\ds 168 \times \map {\sigma_0} {168}\) \(=\) \(\ds 168 \times 16\) $\sigma_0$ of $168$
\(\ds \) \(=\) \(\ds \paren {2^3 \times 3 \times 7} \times 2^4\)
\(\ds \) \(=\) \(\ds 2^7 \times 3 \times 7\)
\(\ds \) \(=\) \(\ds 2688\)


\(\ds 192 \times \map {\sigma_0} {192}\) \(=\) \(\ds 192 \times 14\) $\sigma_0$ of $192$
\(\ds \) \(=\) \(\ds \paren {2^6 \times 3} \times \paren {2 \times 7}\)
\(\ds \) \(=\) \(\ds 2^7 \times 3 \times 7\)
\(\ds \) \(=\) \(\ds 2688\)


\(\ds 224 \times \map {\sigma_0} {224}\) \(=\) \(\ds 224 \times 12\) $\sigma_0$ of $224$
\(\ds \) \(=\) \(\ds \paren {2^5 \times 7} \times \paren {2^2 \times 3}\)
\(\ds \) \(=\) \(\ds 2^7 \times 3 \times 7\)
\(\ds \) \(=\) \(\ds 2688\)



Sources