Amicable Tuples

When I think of amicable numbers, a pair of numbers come to mind: 220 and 284. They have the property that the sum of the proper divisors of 220 equals 284 and vice versa. This is an example of an amicable 2-tuple. The relationship between 220 and 284 can be expressed as:$${\sigma }_1(220)={\sigma }_1(284)=220+284$$This means that any amicable 2-tuple, let's say $(x,y)$, has the property that:$${\sigma }_1(x)={\sigma }_1(y)=x+y$$The next amicable pair or 2-tuple is (1184, 1210). A list of numbers that form amicable pairs is given by OEIS A063990:

A063990

Amicable numbers.

The initial members are:

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, 122265, 122368, 123152, 124155, 139815, 141664, 142310

The sequence lists the amicable numbers in increasing order. Note that the pairs $(x,y)$ are not necessarily adjacent to each other in the list. The first time a pair ordered by its first element is not adjacent is $x$ = 63020, $y$ = 76084 which correspond to a(17) and a(23), respectively.

This leads us on to amicable 3-tuples, $(x,y,z)$, that have the property:$${\sigma }_1(x)={\sigma }_1(y)={\sigma }_1(z)=x+y+z$$The first amicable triple or 3-tuple is (1980, 2016, 2556). My diurnal age today, 27312, is part of the amicable 3-tuple (27312, 21168, 22200).  In general, we can call a finite set $(x_1,x_2,\dots ,x_k)$ of natural numbers (the $x_i$ are pairwise distinct), an amicable $k$-tuple iff$${\sigma }_1(x_1)={\sigma }_1(x_2)=\cdots ={\sigma }_1(x_k)=x_1+x_2+...+x_k$$For $k$=1, the only possible amicable one-tuple is (1).

OEIS A255215 lists numbers that belong to at least one amicable tuple and the initial members are:

1, 220, 284, 1184, 1210, 1980, 2016, 2556, 2620, 2924, 5020, 5564, 6232, 6368, 9180, 9504, 10744, 10856, 11556, 12285, 14595, 17296, 18416, 21168, 22200, 23940, 27312, 31284, 32136, 37380, 38940, 39480, 40068, 40608, 41412, 41952, 42168, 43890, 46368, 47124