An amicable quadruple as a quadruple such that
(1)
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where is the divisor function.
If and are amicable pairs and
(2)
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then is an amicable quadruple. This follows from the identity
(3)
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The smallest known amicable quadruple is (842448600, 936343800, 999426600, 1110817800).
Large amicable quadruples can be generated using the formula
(4)
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where
(5)
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and is a Mersenne prime with a prime (Y. Kohmoto; Guy 1994, p. 59).