Definition:Amicable Pair/Definition 2
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Definition
Let $m \in \Z_{>0}$ and $n \in \Z_{>0}$ be (strictly) positive integers.
$m$ and $n$ are an amicable pair if and only if:
- $\map {\sigma_1} m = \map {\sigma_1} n = m + n$
where $\sigma_1$ denotes the divisor sum function.
Examples
$220$ and $284$
$220$ and $284$ are the smallest amicable pair:
- $\map {\sigma_1} {220} = \map {\sigma_1} {284} = 504 = 220 + 284$
$1184$ and $1210$
$1184$ and $1210$ are the $2$nd amicable pair:
- $\map {\sigma_1} {1184} = \map {\sigma_1} {1210} = 2394 = 1184 + 1210$
$2620$ and $2924$
$2620$ and $2924$ are the $3$rd amicable pair:
- $\map {\sigma_1} {2620} = \map {\sigma_1} {2924} = 5544 = 2620 + 2924$
$5020$ and $5564$
$5020$ and $5564$ are the $4$th amicable pair:
- $\map {\sigma_1} {5020} = \map {\sigma_1} {5564} = 10 \, 584 = 5020 + 5564$
$6232$ and $6368$
$6232$ and $6368$ are the $5$th amicable pair:
- $\map {\sigma_1} {6232} = \map {\sigma_1} {6368} = 12 \, 600 = 6232 + 6368$
$10 \, 744$ and $10 \, 856$
$10 \, 744$ and $10 \, 856$ are the $6$th amicable pair:
- $\map {\sigma_1} {10 \, 744} = \map {\sigma_1} {10 \, 856} = 21 \, 600 = 10 \, 744 + 10 \, 856$
$12 \, 285$ and $14 \, 595$
$12 \, 285$ and $14 \, 595$ are the $7$th amicable pair and the smallest odd amicable pair:
- $\map {\sigma_1} {12 \, 285} = \map {\sigma_1} {14 \, 595} = 26 \, 880 = 12 \, 285 + 14 \, 595$
$17 \, 296$ and $18 \, 416$
$17 \, 296$ and $18 \, 416$ are the $8$th amicable pair:
- $\map {\sigma_1} {17 \, 296} = \map {\sigma_1} {18 \, 416} = 35 \, 712 = 17 \, 296 + 18 \, 416$
Sequence
The sequence of amicable pairs begins:
- $\tuple {220, 284}, \tuple {1184, 1210}, \tuple {2620, 2924}, \tuple {5020, 5564}, \tuple {6232, 6368}, \tuple {10744, 10856}, \tuple {12285, 14595}, \tuple {17296, 18416}, \tuple {63020, 76084}$
Also see
- Results about amicable pairs can be found here.
Sources
- Weisstein, Eric W. "Amicable Pair." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AmicablePair.html