Amicable Quadruple

An amicable quadruple as a quadruple  such that

(1)

where  is the divisor function.

If  and  are amicable pairs and

(2)

then  is an amicable quadruple. This follows from the identity

(3)

The smallest known amicable quadruple is (842448600, 936343800, 999426600, 1110817800).

Large amicable quadruples can be generated using the formula

(4)

where

(5)

and  is a Mersenne prime with  a prime  (Y. Kohmoto; Guy 1994, p. 59).

REFERENCES:

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 59, 1994.