Almost Perfect Numbers Day - August 16

Happy Almost Perfect Numbers Day!

Today we celebrate numbers that aren't perfect numbers but are almost perfect numbers (aka slightly deficient numbers aka least defective numbers). Perfect numbers (in case you've forgotten) are the numbers such that if you sum up their divisors the result is 2 times the number itself. For example, the divisors of 6 are 1, 2, 3, and 6, and if you sum them up you get 1+2+3+6=12, which is also 2 x 6.

Almost perfect numbers are numbers where the sum of the divisors is 1 less than 2 times the number. For example, the divisors of 4 are 1, 2, and 4, and the sum of them is 1+2+4 = 7, which is 1 less than 2 x 4. All the known almost perfect numbers are powers of 2, like 2x2 = 4, 2x2x2 = 8, 2x2x2x2 = 16, etc. It hasn't been proven that every almost perfect number is of this form, though given the computational power we have today, if other forms exist, they must be very large almost perfect numbers.

Here is the beginning of the almost perfect number sequence:



Comments

Post a Comment

Popular posts from this blog

Year In Review - February 2023 thru February 2024

Image
As I've (more or less) successfully managed yet another trip around the sun, so lets discuss what happened. This should be similar to what I did in 2022 and 2023 , so if those bored you nearly comatose you should be warned that this will be more of the same. Photographic evidence that I exist. (Almost certainly not an AI generated image of a random old guy.) Health : It wouldn't be a year in review without a passing mention about my left knee. Although I don't recall it swelling up like it used to do, these days I'll sometimes get a fairly sharp pain if I bend it weirdly (which is harder to avoid than you'd think it would be). Still, I go jogging most mornings ... so how bad can the knee really be? I hurt my back this summer. I attribute this to stupidity. At Duke Farms we were cutting down (mostly invasive) plants that were tangling into or just very close to some of their fencing. Since we were cutting on both sides of the fence but were only collecting the debri
Read more

Telephone Numbers Day - October 26

Image
Happy Telephone Numbers Day! You may be surprised to learn that these aren't the telephone numbers associated with our individual telephones, but instead represent the number of ways that n number of telephones can be connected to each other for 2-way calls. For example, if you have 0 phones or 1 phone, there only 2-way "connection" possible is no connection at all. But if you have 2 phones, there are now 2 possibilities: no connection at all or the 2 phones connected to each other. This table shows the number of combinations possible when you have very few phones: You might notice that the number of combinations starts very small but then grows pretty quickly. This is one reason why it's impractical to set up a telephone network with equipment dedicated to every possible phone call combination. Actually writing and counting all those combinations would get tedious pretty quickly; fortunately there's a formula for that: Applying this formula, you can see just how
Read more

Festival of Wepet Renpet - July 14

Image
The Festival of Wepet Renpet was a matter of life and death in ancient Egypt - it was held each year when the Nile River flooded, irrigating the fields that the Egyptians depended on for much of their food. It was also considered the Egyptian New Year. Since it depended on a flood, the date is an approximation. Of course the ancient Egyptians understood that Wepet Renpet had nothing to do with rain in the Ethiopian highlands; they understood that the deities Khnum and Neith were satisfied with their offerings. Photo by British Library on Unsplash
Read more