Amicable numbers

The Thābit ibn Qurra theorem is a method for discovering amicable numbers invented in the ninth century by the Arab mathematician Thābit ibn Qurra.[5]

It states that if

p = 3×2^{n − 1} − 1,

q = 3×2ⁿ − 1,

r = 9×2^{2n − 1} − 1,

where n > 1 is an integer and p, q, and r are prime numbers, then 2ⁿ×p×q and 2ⁿ×r are a pair of amicable numbers. This formula gives the pairs (220, 284) for n = 2, (17296, 18416) for n = 4, and (9363584, 9437056) for n = 7, but no other such pairs are known. Numbers of the form 3×2ⁿ − 1 are known as Thabit numbers. In order for Ibn Qurra's formula to produce an amicable pair, two consecutive Thabit numbers must be prime; this severely restricts the possible values of n.

To establish the theorem, Thâbit ibn Qurra proved nine lemmas divided into two groups. The first three lemmas deal with the determination of the aliquot parts of a natural integer. The second group of lemmas deals more specifically with the formation of perfect, abundant and deficient numbers.[6]

Euler's rule is a generalization of the Thâbit ibn Qurra theorem. It states that if

p = (2^{n − m} + 1)×2^m − 1,

q = (2^{n − m} + 1)×2ⁿ − 1,

r = (2^{n − m} + 1)²×2^{m + n} − 1,

where n > m > 0 are integers and p, q, and r are prime numbers, then 2ⁿ×p×q and 2ⁿ×r are a pair of amicable numbers. Thābit ibn Qurra's theorem corresponds to the case m = n − 1. Euler's rule creates additional amicable pairs for (m,n) = (1,8), (29,40) with no others being known. Euler (1747 & 1750) overall found 58 new pairs to make all the by then existing pairs into 61.[2][7]

Amicable numbers ${\displaystyle (m,n)}$ satisfy ${\displaystyle \sigma (m)-m=n}$ and ${\displaystyle \sigma (n)-n=m}$ which can be written together as ${\displaystyle \sigma (m)=\sigma (n)=m+n}$. This can be generalized to larger tuples, say ${\displaystyle (n_{1},n_{2},\ldots ,n_{k})}$, where we require

${\displaystyle \sigma (n_{1})=\sigma (n_{2})=\dots =\sigma (n_{k})=n_{1}+n_{2}+\dots +n_{k}}$

For example, (1980, 2016, 2556) is an amicable triple (sequence A125490 in the OEIS), and (3270960, 3361680, 3461040, 3834000) is an amicable quadruple (sequence A036471 in the OEIS).

Amicable multisets are defined analogously and generalizes this a bit further (sequence A259307 in the OEIS).

Sociable numbers are the numbers in cyclic lists of numbers (with a length greater than 2) where each number is the sum of the proper divisors of the preceding number. For example, ${\displaystyle 1264460\mapsto 1547860\mapsto 1727636\mapsto 1305184\mapsto 1264460\mapsto \dots }$ are sociable numbers of order 4.

The aliquot sequence can be represented as a directed graph, ${\displaystyle G_{n,s}}$, for a given integer ${\displaystyle n}$, where ${\displaystyle s(k)}$ denotes the sum of the proper divisors of ${\displaystyle k}$.[10] Cycles in ${\displaystyle G_{n,s}}$ represent sociable numbers within the interval ${\displaystyle [1,n]}$. Two special cases are loops that represent perfect numbers and cycles of length two that represent amicable pairs.