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 divisor of these numbers is 1, so that σ(n)/n is an irreducible fraction – then the number n is solitary . For a prime number p we have σ(p) = p + 1, which is co-prime with p. No general method is known for determining whether a number is "friendly" or solitary. The smallest number whose classification is unknown is 10; it is conjectured to be solitary. If it is not, its smallest friend is at least . Small numbers with a relatively large smallest friend do exist: for instance, 24 is "friendly", with its smallest friend 91,963,648. Large clubs It is an open problem whether there are infinitely large clubs of mutually "friendly" numbers. The perfect numbers form a club, and it is conjectured that there are infinitely many perfect numbers (at least as many as there are Mersenne primes), but no proof is known. , 51 perfect numbers are known, the largest of which has more than 49 million digits in decimal notation. There are clubs with more known members: in particular, those formed by multiply perfect numbers, which are numbers whose "abundancy" is an integer. , the club of "friendly" numbers with "abundancy" equal to 9 has 2130 known members. Although some are known to be quite large, clubs of multiply perfect numbers (excluding the perfect numbers themselves) are conjectured to be finite. Asymptotic density Every pair a, b of friendly numbers gives rise to a positive proportion of all natural numbers being friendly (but in different clubs), by considering pairs na, nb for multipliers n with gcd(n, ab) = 1. For example, the "primitive" friendly pair 6 and 28 gives rise to friendly pairs 6n and 28n for all n that are congruent to 1, 5, 11, 13, 17, 19, 23, 25, 29, 31, 37, or