Definition:Amicable Pair/Definition 2

From ProofWiki
Jump to navigation Jump to search

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:

$\map \sigma m = \map \sigma n = m + n$

where $\map \sigma m$ denotes the $\sigma$ function.


Examples

$220$ and $284$

$220$ and $284$ are the smallest amicable pair:

$\map \sigma {220} = \map \sigma {284} = 504 = 220 + 284$


$1184$ and $1210$

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

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


$2620$ and $2924$

$2620$ and $2924$ are the $3$rd amicable pair:

$\map \sigma {2620} = \map \sigma {2924} = 5544 = 2620 + 2924$


$5020$ and $5564$

$5020$ and $5564$ are the $4$th amicable pair:

$\map \sigma {5020} = \map \sigma {5564} = 10 \, 584 = 5020 + 5564$


$6232$ and $6368$

$6232$ and $6368$ are the $5$th amicable pair:

$\sigma \left({6232}\right) = \sigma \left({6368}\right) = 12 \, 600 = 6232 + 6368$


$10 \, 744$ and $10 \, 856$

$10 \, 744$ and $10 \, 856$ are the $6$th amicable pair:

$\sigma \left({10 \, 744}\right) = \sigma \left({10 \, 856}\right) = 21 \, 600 = 10 \, 744 + 10 \, 856$


$12 \, 285$ and $14 \, 595$

$12 \, 285$ and $14 \, 595$ are the $7$th amicable pair and the smallest odd amicable pair:

$\map \sigma {12 \, 285} = \map \sigma {14 \, 595} = 26 \, 880 = 12 \, 285 + 14 \, 595$


$17 \, 296$ and $18 \, 416$

$17 \, 296$ and $18 \, 416$ are the $8$th amicable pair:

$\map \sigma {17 \, 296} = \map \sigma {18 \, 416} = 35 \, 712 = 17 \, 296 + 18 \, 416$


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