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.
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 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 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. There are several unsolved problems related to the friendly numbers.