What is is 21 and 18 weird?

In the context of mathematics, particularly number theory, a "weird number" is defined as a natural number that is abundant but not semiperfect. Let's break these terms down:

1. Abundant Number: A natural number is considered abundant if the sum of its proper divisors (divisors excluding the number itself) is greater than the number. For example, the number 12 is abundant because the proper divisors of 12 (1, 2, 3, 4, and 6) sum to 16, which is greater than 12.

2. Semiperfect Number: A natural number is semiperfect if it equals the sum of some (or all) of its proper divisors. Semiperfect numbers include perfect numbers and numbers that can be expressed as the sum of some of their divisors.

3. Weird Number: A natural number that is abundant but not semiperfect. This means that while the sum of the proper divisors is greater than the number itself (making it abundant), no subset of these divisors adds up to the number itself.

Now, examining the numbers 21 and 18:

• For 21: Its proper divisors are 1, 3, and 7, which sum to 11. Since 11 is less than 21, 21 is not abundant, and therefore cannot be weird.

• For 18: Its proper divisors are 1, 2, 3, 6, and 9, which sum to 21. Since 21 is greater than 18, 18 is abundant. However, 18 can be expressed as the sum of some of its divisors (9 + 6 + 3 = 18), making it semiperfect. Therefore, 18 is not weird.

In conclusion, neither 21 nor 18 is a weird number.