Friendly Pair

Define the abundancy  of a positive integer  as

(1)

where  is the divisor function. Then a pair of distinct numbers  is a friendly pair (and  is said to be a friend of ) if their abundancies are equal:

(2)

For example, (4320, 4680) is a friendly pair since , , and

			(3)
			(4)

Another example is , which has index 5/2. The first few friendly pairs, ordered by smallest maximum element are (6, 28), (30, 140), (80, 200), (40, 224), (12, 234), (84, 270), (66, 308), ... (OEIS A050972 and A050973).

Friendly triples and higher-order tuples are also possible. Friendly triples include (2160, 5400, 13104), (9360, 21600, 23400), and (4320, 4680, 26208), friendly quadruples include (6, 28, 496, 8128), (3612, 11610, 63984, 70434), (3948, 12690, 69936, 76986), and friendly quintuples include (84, 270, 1488, 1638, 24384), (30, 140, 2480, 6200, 40640), (420, 7440, 8190, 18600, 121920).

Numbers that have friends are called friendly numbers, and numbers that do not have friends are called solitary numbers. A sufficient (but not necessary) condition for  to be a solitary number is that , where  is the greatest common divisor of  and . There are some numbers that can easily be proved to be solitary, but the status of numbers 10, 14, 15, 20, and many others remains unknown (Hickerson 2002).

Hoffman (1998, p. 45) uses the term "friendly numbers" to describe amicable pairs.

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.

Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. New York: Hyperion, 1998.

Pollack, P. and Pomerance, C. "Some Problems of Erdős on the Sum-of-Divisors Function." Trans. Amer. Math. Soc. 3, 1-26, 2016.

Sloane, N. J. A. Sequences A050972 and A050973 in "The On-Line Encyclopedia of Integer Sequences."