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 $224$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {2^5 \times 7} \times \paren {2^2 \times 3}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^7 \times 3 \times 7\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2688\) |
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Sources
- 1994: Richard K. Guy: Unsolved Problems in Number Theory (2nd ed.)
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $168$