Odd Amicable Pair/Examples/1,175,265-1,438,983
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Example of Odd Amicable Pair
$1 \, 175 \, 265$ and $1 \, 438 \, 983$ are the $9$th odd amicable pair:
- $\map {\sigma_1} {1 \, 175 \, 265} = \map {\sigma_1} {1 \, 438 \, 983} = 2 \, 614 \, 240 = 1 \, 175 \, 265 + 1 \, 438 \, 983$
Proof
By definition, $m$ and $n$ form an amicable pair if and only if:
- $\map {\sigma_1} m = \map {\sigma_1} n = m + n$
where $\map {\sigma_1} n$ denotes the divisor sum function.
Thus:
\(\ds \map {\sigma_1} {1 \, 175 \, 265}\) | \(=\) | \(\ds 2 \, 614 \, 240\) | $\sigma_1$ of $1 \, 175 \, 265$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 1 \, 175 \, 265 + 1 \, 438 \, 983\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map {\sigma_1} {1 \, 438 \, 983}\) | $\sigma_1$ of $1 \, 438 \, 983$ |
$\blacksquare$
Historical Note
The odd amicable pair $1 \, 175 \, 265$ and $1 \, 438 \, 983$ was discovered by G.W. Kraft in the $17$th century.
It was the $1$st odd amicable pair to be discovered.
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $1,175,265$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1,175,265$