Is 10 a solitary number?

A number which does not have anyfriends. Solitary numbers include allprimes,prime powers, and numbers for which, whereis thegreatest common divisorofandandis thedivisor function. The first few numbers satisfyingare 1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 21, ... 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

Some numbers can be proved not to be solitary by finding another integer with the same index, although sometimes the smallest such number is fairly large. For example, 24 isfriendlybecauseis afriendly pair. However, there exist numbers such as, 45, 48, and 52 which are solitary but for which. 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.

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Shaik Aasif

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