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. 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.

The numbers 1 through 5 are all solitary. The smallest friendly number is 6, forming for example, the friendly pair 6 and 28 with abundancy σ(6) / 6 = (1+2+3+6) / 6 = 2, the same as σ(28) / 28 = (1+2+4+7+14+28) / 28 = 2. The shared value 2 is an integer in this case but not in many other cases. Numbers with abundancy 2 are also known as perfect numbers. There are several unsolved problems related to the friendly numbers.

In spite of the similarity in name, there is no specific relationship between the friendly numbers and the amicable numbers or the sociable numbers, although the definitions of the latter two also involve the divisor function.