Amicable triple

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In mathematics, an amicable triple is a set of three different numbers so related that the restricted sum of the divisors of each is equal to the sum of other two numbers. [1] [2]

In another equivalent characterization, an amicable triple is a set of three different numbers so related that the sum of the divisors of each is equal to the sum of the three numbers.

So a triple (a, b, c) of natural numbers is called amicable if s(a) = b+c, s(b) = a+c and s(c) = a+b, or equivalently if σ(a) = σ(b) = σ(c) = a+b+c. Here σ(n) is the sum of all positive divisors, and s(n) = σ(n) − n is the aliquot sum. The smallest amicable triple is (1980, 2016, 2556). [3]

References

  1. Dickson, L. E. (1913-03-01). "Amicable Number Triples" . The American Mathematical Monthly. 20 (3): 84–92. doi:10.1080/00029890.1913.11997926. ISSN   0002-9890.
  2. Dickson, L. E. (1913). "Amicable Number Triples" . The American Mathematical Monthly. 20 (3): 84–92. doi:10.2307/2973442. ISSN   0002-9890. JSTOR   2973442.
  3. Mason, Thomas E. (1921). "On Amicable Numbers and Their Generalizations" . The American Mathematical Monthly. 28 (5): 195–200. doi:10.2307/2973750. ISSN   0002-9890. JSTOR   2973750.