Amenable number

Short description: Type of positive integer

An amenable number is a positive integer for which there exists a multiset of as many integers as the original number that both add up to the original number and when multiplied together give the original number. To put it algebraically, for a positive integer n, there is a multiset of n integers {a₁, ..., a_n}, for which the equalities $$n=\sum _{i=1}^n{a}_i=\prod _{i=1}^n{a}_i$$ hold.

The amenable numbers were named in a 1995 problem posed by H. Tamvakis. Whether a number is amenable can be determined from its value modulo 4.

Examples

For example, 5 is amenable since 5 = 1 + (−1) + 1 + (−1) + 5.

The first few amenable numbers are

1, 5, 8, 9, 12, 13 ... (sequence A100832 in the OEIS).

Characterization

the amenable numbers consist exactly of the positive integers that are congruent to 0 or 1 modulo 4, and are not equal to 4.

For the integers of the form $n=4k+1$, where $k$ is itself an integer, a solution can be given by a set of $2k$ copies of $+1$, $2k$ copies of $-1$, and one copy of $n$, generalizing the example of 5 given above. For the integers of the form $n=4k$, with $k>1$, a solution can be given by $n-2$ copies of $\pm 1$ and one copy each of $\pm 2$ and $2k$.

Four can be shown not to be amenable by case analysis. For numbers $n$ congruent to 2 mod 4, it is only possible for the product of $n$ integers to equal $n$ when there is a single even term, from which it follows that the sum of the terms is odd and therefore unequal to $n$. When $n$ is congruent to 3 mod 4, all terms must be odd and there must be an odd number of terms congruent to 3 mod 4, from which it follows that the sum is 1 mod 4 and unequal to $n$.

Other properties

It follows from their characterization that the set of amenable numbers is closed under multiplication: the product of two amenable numbers is an amenable number.

Related concepts

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