A solitary number is a number which does not have any friends. Solitary numbers include all primes, prime powers, and numbers for which (n, σ(n)) = 1, where (a, b) is the greatest common divisor of a and b and σ(n) is the divisor function. The first few numbers satisfying (n, σ(n)) = 1 are 1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 21, ... (OEIS A014567). Numbers such as 18, 45, 48, 52, 136, 148, 160, 162, 176, 192, 196, 208, 232, 244, 261, 272, 292, 296, 297, 304, 320, 352, and 369 can also be easily proved to be solitary.

friend | friendly number | friendly pair | prime power