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

Superb Owl Sunday - First Sunday in February

Image
Note : Superb Owl Sunday is the 2nd Sunday in February starting in 2023. Today is Superb Owl Sunday! It's a day to celebrate how superb owls are. (Apparently a football league has repositioned a couple letters as the inspiration for their championship game , but obviously this is of secondary importance to owls.) Most owls are the "night shift" of predatory birds, though it's not unheard of for some to be active in daylight. A Snowy Owl that's visited Central Park was definitely a superb owl. Here are a couple education owls residing at The Raptor Trust in central New Jersey. Red-morph Eastern Screech Owl - The Raptor Trust Barred Owl - The Raptor Trust A former bird rehabber a little further away, the Delaware Valley Raptor Center , also has superb owls living with them. Saw-whet Owl - Delaware Valley Raptor Center Great Horned Owl - Delaware Valley Raptor Center
Read more

Day Without Mudville Joy - June 3

Image
On this date in 1888, Ernest Thayer's poem Casey at the Bat was first published in The San Francisco Examiner . The outlook wasn't brilliant for the Mudville nine that day: The score stood four to two, with but one inning more to play, And then when Cooney died at first, and Barrows did the same, A pall-like silence fell upon the patrons of the game. A straggling few got up to go in deep despair. The rest Clung to the hope which springs eternal in the human breast; They thought, "If only Casey could but get a whack at that— We'd put up even money now, with Casey at the bat." But Flynn preceded Casey, as did also Jimmy Blake, And the former was a hoodoo, while the latter was a cake; So upon that stricken multitude grim melancholy sat, For there seemed but little chance of Casey getting to the bat. But Flynn let drive a single, to the wonderment of all, And Blake, the much despisèd, tore the cover off the ball; And when the dust had lifted, and men saw what had oc...
Read more