Solitary Number

A solitary number is a number which does not have any friends. Solitary numbers include all primes, prime powers, and numbers for which , where  is the greatest common divisor of  and  and  is the divisor function. The first few numbers satisfying  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 (Hickerson 2002).

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 is friendly because  is a friendly 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.

In 1996, Carl Pomerance told Dean Hickerson that he could prove that the solitary numbers have positive density, thus disproving a conjecture by Anderson and Hickerson (1977). However, this proof was never published (Hickerson 2002), and Pomerance has since been unable to reproduce it, leading to retraction of the claimed proof (C. Pomerance, pers. comm., Dec. 28, 2016).

REFERENCES:

Anderson, C. W. and Hickerson, D. Problem 6020. "Friendly Integers." Amer. Math. Monthly 84, 65-66, 1977.

Hickerson, D. "Re: friendly/solitary numbers [was: typos]" seqfan@ext.jussieu.fr mailing list. 19 Sep 2002.

Sloane, N. J. A. Sequence A014567 in "The On-Line Encyclopedia of Integer Sequences."