What is is 18 and 21 weird?

In mathematics, a "weird number" is defined as a natural number that is abundant but not semiperfect. To clarify these terms:

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

2. Semiperfect Number: A number is semiperfect if it is equal to the sum of a subset of its proper divisors. If at least one subset of the divisors sums to the number itself, the number is semiperfect.

A number can be weird if it is abundant but no subset of its divisors can sum back to the number itself.

Now, let's look at the numbers 18 and 21:

• For 18: The proper divisors are 1, 2, 3, 6, and 9, which sum to 21. Since 21 is greater than 18, 18 is abundant. However, 18 is semiperfect because one of its subsets of divisors—1, 2, 3, 6, and 6—sums to 18. Therefore, 18 is not a weird number.

• For 21: The proper divisors are 1, 3, and 7, which sum to 11. Since 11 is less than 21, 21 is not an abundant number. Thus, it cannot be a weird number, as a number must first be abundant before it is considered for being "weird."

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