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$$ \sigma_1(x_1)= \sigma_1(x_2)=\dots =\sigma_1(x_k)=x_1+x_2+...+x_k $$For \(k\)=1, the only possible amicable one-tuple is (1).
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
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