odditudes-and-weirdities

Happy Semiperfect Numbers Day!  While perfect numbers are numbers that are equal to the sum of all their proper divisors, semiperfect numbers are equal to the sum of some of their proper divisors. (Yes, this means that all perfect numbers are also semiperfect numbers.) Similar to perfect numbers, semiperfect numbers don't seem to have a lot of application but are more of numerical oddities. Here are the smallest semiperfect numbers, including a couple that are also perfect numbers: And here's the arithmetic that shows why the smaller numbers are or are not semiperfect numbers. We celebrate Semiperfect Numbers Day on December 18 because 12/18 is the latest (American style) date that can be formed using consecutive semiperfect numbers.

Happy Deficient Numbers Day! Deficient numbers (aka defective numbers) are the somewhat disdainful name for the natural numbers that are the companions for the more prestigiously named perfect numbers and abundant numbers . They are numbers where the sum of their divisors is less than twice that number. The start of the deficient numbers is below. It's known that there are an infinite number of both even and odd deficient numbers. Though most odd numbers are deficient, there are exceptions; 945 is the smallest odd abundant number, and thus is the first odd number that's not deficient. Here is the classification the beginning of the natural numbers into deficient, abundant, and perfect numbers. (Some are also noted as semiperfect or almost perfect numbers .) We celebrate the deficient numbers today because the highest 2 consecutive deficient numbers that can form a date are 11 and 13: 11/13 or November 13.