Friendly number

In number theory, friendly numbers are two or more natural numbers with a common abundancy index, 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.

Being mutually friendly is an equivalence relation, and thus induces a partition of the positive naturals into clubs (equivalence classes) of mutually friendly numbers.

A number that is not part of any friendly pair is called solitary.

The abundancy index 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 ${\displaystyle \sigma _{-\!1}(n)}$ where ${\displaystyle \sigma _{k}}$ denotes a divisor function with ${\displaystyle \sigma _{k}(n)}$ equal to the sum of the k-th powers of the divisors of n.

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