Solitary number

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In mathematics a solitary number is number which does not have any "friends".

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.

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.