||This article relies largely or entirely upon a single source.
(November 2011)| In number theory, friendly numbers are two or
more natural numbers with a common abundancy, the ratio between
the sum of divisors of a number and the number itself. Two
numbers with the same abundancy form a friendly pair; n numbers
with the same abundancy form a friendly n-tuple. A number that
is not part of any friendly pair is called solitary. The
abundancy of n is the rational number σ(n) / n, in which σ
denotes the sum of divisors function. A number n is a friendly
number if there exists m ≠ n such that σ(m) / m = σ(n) / n. Note
that abundancy is not the same as abundance which is defined as
σ(n) − 2n. Abundancy may also be expressed as where denotes a
divisor function with equal to the sum of the k-th powers of the
divisors of n. The numbers 1 through 5 are all solitary. The
smallest friendly number is 6, forming for example the friendly
pair (6, 28) with abundancy σ(6) / 6 = (1+2+3+6) / 6 = 2, the
same as σ(28) / 28 = (1+2