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Is 10 a solitary number?

prashant kashya , 10 Years ago
Grade upto college level
anser 1 Answers
SHAIK AASIF AHAMED

Last Activity: 10 Years ago

Hello student,

A number which does not have anyfriends. Solitary numbers include allprimes,prime powers, and numbers for which(n,sigma(n))=1, where(a,b)is thegreatest common divisorofaandbandsigma(n)is thedivisor function. The first few numbers satisfying(n,sigma(n))=1are 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 isfriendlybecause(24,91963648)is afriendly pair. However, there exist numbers such asn=18, 45, 48, and 52 which are solitary but for which(n,sigma(n))!=1. 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.

Thanks and Regards

Shaik Aasif

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