Friendly Pair

Define the abundancy Sigma(n) of a positive integer n as


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


For example, (4320, 4680) is a friendly pair since sigma(4320)=15120, sigma(4680)=16380, and


Another example is (24,91963648), 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 n to be a solitary number is that (sigma(n),n)=1, where (a,b) is the greatest common divisor of a and b. 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.

See also

Abundancy, Aliquot Sequence, Amicable Pair, Friend, Friendly Number, Solitary Number

Explore with Wolfram|Alpha


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]" 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."

Referenced on Wolfram|Alpha

Friendly Pair

Cite this as:

Weisstein, Eric W. "Friendly Pair." From MathWorld--A Wolfram Web Resource.

Subject classifications