In mathematics a solitary number is number which does not have any "friends". Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ...

Two numbers m and n are friends if and only if σ(m)/m = σ(n)/n. Then, it is said that (m, n) is a friendly pair.

All numbers for which (n, σ(n)) = 1 are solitary, where (a, b) is the greatest common divisor of a and b, and σ(n) is the divisor function. This implies that all primes and prime powers are solitary. In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf), of two non-zero integers, is the largest positive integer that divides both numbers. ... Divisor function σ0(n) up to n=250 Sigma function σ1(n) up to n=250 Sum of the squares of divisors, σ2(n), up to n=250 Sum of cubes of divisors, σ3(n) up to n=250 In mathematics, and specifically in number theory, a divisor function is... In mathematics, a prime number, or prime for short, is a natural number greater than one and whose only distinct positive divisors are 1 and itself. ... In mathematics, a prime power is a positive integer power of a prime number. ...

The first few numbers which satisfy (n, σ(n)) = 1, are 1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 21, ...

Numbers can be proven not to be solitary by finding another integer which is a friend, although sometimes the smallest such number is fairly large. For example, 24 is not solitary because (24, 91963648) is a friendly pair. However, there exist numbers which are solitary but do not statisfy (n, σ(n)) = 1; they are 18, 45, 48, and 52. It is believed that 10, 14, 15, 20, 22, 26, 33, 34, 38, 44, 46, 51, 54, 58, 62, 68, 69, 70, 72, 74, 76, 82, 86, 87, 88, 90, 91, 92, 94, 95, 99, 104, 105, 106, and many others are also solitary, although a proof appears to be extremely difficult.