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).