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# Amicable numbers

Two integers ${\displaystyle \scriptstyle m\,}$ and ${\displaystyle \scriptstyle n\,}$ are amicable numbers (an amicable pair of numbers) if ${\displaystyle \scriptstyle \sigma (m)-m\,=\,n\,}$ and ${\displaystyle \scriptstyle \sigma (n)-n\,=\,m,\,}$ where ${\displaystyle \scriptstyle \sigma (n)\,}$ is the sum of divisors of ${\displaystyle \scriptstyle n\,}$.

The larger numbers of amicable pairs are of course deficient numbers, while the smallest numbers of amicable pairs are abundant numbers. One might say that the pair is mutually perfect (so to speak) since the abundancy of the smaller number cancels out the deficiency of the larger number, i.e.

${\displaystyle (\sigma (m)+\sigma (n))-2(m+n)=0.\,}$

The amicable pairs are

{(220, 284), (1184, 1210), (2620, 2924), (5020, 5564), (6232, 6368), (10744, 10856), (12285, 14595), (17296, 18416), (63020, 76084), (66928, 66992), (67095, 71145), (69615, 87633), ...}

## Sequences

A063990 Amicable numbers. (sorted union of A002025 and A002046)

{220, 284, 1184, 1210, 2620, 2924, 5020, 5564, 6232, 6368, 10744, 10856, 12285, 14595, 17296, 18416, 63020, 66928, 66992, 67095, 69615, 71145, 76084, 79750, 87633, 88730, 100485, ...}

A002025 Smaller of amicable pair.

{220, 1184, 2620, 5020, 6232, 10744, 12285, 17296, 63020, 66928, 67095, 69615, 79750, 100485, 122265, 122368, 141664, 142310, 171856, 176272, 185368, 196724, 280540, 308620, ...}

A002046 Larger of amicable pair.

{284, 1210, 2924, 5564, 6368, 10856, 14595, 18416, 76084, 66992, 71145, 87633, 88730, 124155, 139815, 123152, 153176, 168730, 176336, 180848, 203432, 202444, 365084, 389924, ...}

A066539 Difference between larger and smaller terms of n-th amicable pair.

{64, 26, 304, 544, 136, 112, 2310, 1120, 13064, 64, 4050, 18018, 8980, 23670, 17550, 784, 11512, 26420, 4480, 4576, 18064, 5720, 84544, 81304, 110852, 43184, 17888, 17150, 11680, ...}

• Perfect numbers (singles) (${\displaystyle \scriptstyle \sigma (n)-n\,=\,n\,}$)
• Amicable numbers (pairs) (${\displaystyle \scriptstyle \sigma (m)-m\,=\,n\,}$ and ${\displaystyle \scriptstyle \sigma (n)-n\,=\,m\,}$)
• Sociable numbers (${\displaystyle \scriptstyle k\,}$-tuples) (${\displaystyle \scriptstyle \sigma (n_{i})-n_{i}\,=\,n_{(i+1{\bmod {k}})},\,i\,=\,0..k-1,\,k\,\geq \,3\,}$)