amenable number

An amenable number is an integer for which there exists a multiset of at least two positive integers that can be either added up or multiplied together to give the original number. To put it algebraically, for an integer n, there is a set of integers a1,,ak, for which the equalities


hold. An elegant example is the number 6, which is an amenable number since 6=1+2+3=123.

All composite numbersMathworldPlanetmath are amenable because, even if other solutions are available, one can always make an inelegant solution by taking the prime factorizationMathworldPlanetmath (expressed with repeated factors rather than exponents) and add as many 1s as necessary to add up to n. Because of the multiplicative identityPlanetmathPlanetmath, multiplying this set of integers will yield n no matter how many 1s there are in the set.

If we allow negative integers in the set, then all prime numbersMathworldPlanetmath are also amenable. Even if no other solutions are available, one can always make an inelegant solution for a prime number p of 1,-1, 1,-1,p. In the sum, the two positive ones are cancelled out by the two negative ones, leaving p, while in the product, the two negative ones cancel out the effect of their signs, leaving only the multiplicative identity to affect the final result.

Amenable numbers should not be confused with amicable numbers, which are pairs of integers whose divisorsMathworldPlanetmathPlanetmath add up to each other.

Title amenable number
Canonical name AmenableNumber
Date of creation 2013-03-22 16:37:02
Last modified on 2013-03-22 16:37:02
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 5
Author PrimeFan (13766)
Entry type Definition
Classification msc 11B34