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 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. 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. Examples As another example, 30 and 140 form a friendly pair, because 30 and 140 have the same "abundancy": The numbers 2480, 6200 and 40640 are also members of this club, as they each have an "abundancy" equal to 12/5. For an example of odd numbers being friendly, consider 135 and 819 ("abundancy" 16/9 (deficient)). There are also cases of even being "friendly" to odd, such as 42 and 544635 ("abundancy" 16/7). The odd "friend" may be less than the even one, as in 84729645 and 155315394 ("abundancy" 896/351). A square number can be friendly, for instance both 693479556 (the square of 26334) and 8640 have "abundancy" 127/36 (this example is accredited to Dean Hickerson). Status for small n In the table below, blue numbers are proven friendly , red numbers are proven solitary , numbers n such that n and are coprime are left uncolored, though they are known to be solitary. Other numbers have unknown status and are yellow. Solitary numbers A number that belongs to a singleton club, because no other number is "friendly" with it, is a solitary number. All prime numbers are known to be solitary, as are powers of prime numbers. More generally, if the numbers n and σ(n) are coprime – meaning that the greatest common