Pairs 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 following pairs of integers $T$ have the property that $m \, \map {\sigma_0} m$ is equal for each $m \in T$:

$\set {18, 27}$
$\set {24, 32}$
$\set {56, 64}$


Proof

\(\ds 18 \times \map {\sigma_0} {18}\) \(=\) \(\ds 18 \times 6\) $\sigma_0$ of $18$
\(\ds \) \(=\) \(\ds \paren {2 \times 3^2} \times \paren {2 \times 3}\)
\(\ds \) \(=\) \(\ds 2^2 \times 3^3\)
\(\ds \) \(=\) \(\ds 108\)
\(\ds 27 \times \map {\sigma_0} {27}\) \(=\) \(\ds 27 \times 4\) $\sigma_0$ of $27$
\(\ds \) \(=\) \(\ds 2^2 \times 3^3\)
\(\ds \) \(=\) \(\ds 108\)


\(\ds 24 \times \map {\sigma_0} {24}\) \(=\) \(\ds 24 \times 8\) $\sigma_0$ of $24$
\(\ds \) \(=\) \(\ds \paren {2^3 \times 3} \times 2^3\)
\(\ds \) \(=\) \(\ds 2^6 \times 3\)
\(\ds \) \(=\) \(\ds 192\)
\(\ds 32 \times \map {\sigma_0} {32}\) \(=\) \(\ds 32 \times 6\) $\sigma_0$ of $32$
\(\ds \) \(=\) \(\ds 2^5 \times \paren {2 \times 3}\)
\(\ds \) \(=\) \(\ds 192\)


\(\ds 56 \times \map {\sigma_0} {56}\) \(=\) \(\ds 56 \times 8\) $\sigma_0$ of $56$
\(\ds \) \(=\) \(\ds \paren {2^3 \times 7} \times 2^3\)
\(\ds \) \(=\) \(\ds 2^6 \times 7\)
\(\ds \) \(=\) \(\ds 448\)
\(\ds 64 \times \map {\sigma_0} {64}\) \(=\) \(\ds 64 \times 7\) $\sigma_0$ of $64$
\(\ds \) \(=\) \(\ds 2^6 \times 7\)
\(\ds \) \(=\) \(\ds 448\)

$\blacksquare$


Sources

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