Amenable number
Encyclopedia
An amenable number is an integer
for which there exists a multiset
of as many integers as the original number 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 multiset of n integers {a1, ... , an}, for which the equalities
hold. Negative numbers are allowed in the multiset. For example, 5 is amenable since 5 = 1 + (-1) + 1 + (-1) + 5. All numbers congruent to 0 or 1, mod 4, except 4, are amenable.
All composite number
s would be amenable if the multiset was allowed to be of any length, because, even if other solutions are available, one can always make an inelegant solution by taking the prime factorization (expressed with repeated factors rather than exponents) and add as many 1s as necessary to add up to n. Because of the multiplicative identity, multiplying this set of integers will yield n no matter how many 1s there are in the set.
To allow any prime number
to be amenable, negative numbers are usually required. 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 positive ones are cancelled out by the 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 number
s, which are pairs of integers whose divisors add up to each other.
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
for which there exists a multiset
Multiset
In mathematics, the notion of multiset is a generalization of the notion of set in which members are allowed to appear more than once...
of as many integers as the original number 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 multiset of n integers {a1, ... , an}, for which the equalities
hold. Negative numbers are allowed in the multiset. For example, 5 is amenable since 5 = 1 + (-1) + 1 + (-1) + 5. All numbers congruent to 0 or 1, mod 4, except 4, are amenable.
All composite number
Composite number
A composite number is a positive integer which has a positive divisor other than one or itself. In other words a composite number is any positive integer greater than one that is not a prime number....
s would be amenable if the multiset was allowed to be of any length, because, even if other solutions are available, one can always make an inelegant solution by taking the prime factorization (expressed with repeated factors rather than exponents) and add as many 1s as necessary to add up to n. Because of the multiplicative identity, multiplying this set of integers will yield n no matter how many 1s there are in the set.
To allow any prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
to be amenable, negative numbers are usually required. 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 positive ones are cancelled out by the 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 number
Amicable number
Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other number. A pair of amicable numbers constitutes an aliquot sequence of period 2...
s, which are pairs of integers whose divisors add up to each other.