Amicable Pair/Examples/1184-1210

Example of Amicable Pair

$1184$ and $1210$ are the $2$nd amicable pair:

$\map {\sigma_1} {1184} = \map {\sigma_1} {1210} = 2394 = 1184 + 1210$

Proof

Let $\map s n$ denote the aliquot sum of $n$.

By definition:

$\map s n = \map {\sigma_1} n - n$

where $\sigma_1$ denotes the divisor sum function.

Thus:

\(\ds \map s {1184}\)

\(=\)

\(\ds \map {\sigma_1} {1184} - 1184\)

\(\ds \)

\(=\)

\(\ds 2394 - 1184\)

$\sigma_1$ of $1184$

\(\ds \)

\(=\)

\(\ds 1210\)

\(\ds \map s {1210}\)

\(=\)

\(\ds \map {\sigma_1} {1210} - 1210\)

\(\ds \)

\(=\)

\(\ds 2394 - 1210\)

$\sigma_1$ of $1210$

\(\ds \)

\(=\)

\(\ds 1184\)

$\blacksquare$

Historical Note

The amicable pair $1184$ and $1210$ was discovered by Nicolò Paganini in $1866$, at the age of $16$.

It is remarkable that it had until that time escaped being found, even by Leonhard Paul Euler's systematic exploration.

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $1184$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1184$

• Weisstein, Eric W. "Amicable Pair." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AmicablePair.html