The Six Triperfect Numbers - Numberphile
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- Опубликовано: 29 сен 2024
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This video features Dr James Grime - singingbanana.com
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Numberphile use semicolons to separate the numbers 😝
I found 2 sistems of formulas that describes all the primes except 2 and 3, could you help me to publish it ?
Gamer_ Obscuro make a blog and put it there. Maybe send it to a journal as well, and once it’s published maybe numberphile will do a video on you
This is not secure, can be stolen from me...
Gamer_ Obscuro I know don’t put here. I said get it in a journal or sonething
It's really nice how the number of triperfect numbers is a perfect number.
Gle fakat
Its just perfect
Thanks, I did not even notice :o
It would be better though if all the other numbers of x-perfect numbers were a y-perfect number as well though
And it is also the smallest perfect number.
The perfect number doesn't exs......
*f o u r t y s i x m i l l i o n*
@@Magnus_Deus Digits long*
2
??
6
As Dr. Grime started listing the other N-perfect numbers (4-perfect, 5-perfect, 6-perfect and 7-perfect), and noticing that above N=2 (the perfect numbers), all of them are thought to be finite, I wondered if there's also a ceiling to N.
But then again, there seems to be no pattern to the numbers of N-perfect numbers.
Sorry, just throwing that thought out there.
-------------------
EDIT:
And just as I've suspected, there's already investigation along these lines.
en.wikipedia.org/wiki/Multiply_perfect_number
I wonder how it would look if you graphed all the triperfect numbers on a graph
Cool! Does the amount of discovered n-perfect numbers increase as n increases? That seemed to be the case with these examples
Daniel Gonzalez We should look at
S(n) = the sum of factors of n
Maybe take an average of it or the max and study the divergence
boumbh interesting, thanks!
You would need to remove the word "discovered" for there to be a hope of that being true.
Hey James iam sure u can. Do everything to know prime distrubution. Everythin everything
perfect
So is he listing these numbers from memory, or is there an off screen list he's referencing?
Remembering that 1 is not a prime:
Are there any composite numbers whose primefactors add up to the number itself?
Lau Bjerno If we ignore the multiplicity of the prime factors in the prime factorization of the number, then no, no such number exists. In fact, the sum will always be much less than the number. This is because multiplication grows much faster than summation.
If we include the multiplicity of the factor, then the answer is yes. For example, 4 has the prime factorization 2^2, and 2 + 2 = 2*2 = 4. Other than this, though, no power of a prime satisfies this criteria. Trivially, no square-fre semi-prime composite satisfies it either. In fact, I believe 4 is the only number to satisfy this criteria.
So wait... how do you know that if there is an odd-perfect number, than double that would be a tri-perfect number? I'd like to hear the explanation on that
If there is an odd perfect number, n, then all of its divisors (including itself) sum to 2n. call these divisors d0, d1, d2...etc. so d0+d1+d2+... = 2n.
The proposed triperfect number, 2n, would have all the same divisors as n but in addition, it would have all the doubles of the divisors ie. the divisors of 2n would be d0, d1, d2... & 2d0, 2d1, 2d2....
If you add all of the divisors of 2n, you would get d0+d1+d2+...+2d0+2d1+2d2... = 3d0+3d1+3d2... = 3(d0+d1+d2...) = 3 x 2n (because d0+d1+d2...=2n) = 6n. So the sum of the divisors of 2n would be 6n therefore 2n would be a triperfect number.
If n is a triperfect number that is not divisible by 3, then 3n is a 4-perfect number!
"The largest one we've found is 46 million..."
That's not so bad. I could check larger numbers than that easily.
"Digits long."
Nevermind
hey, depending on the calculus you want to make 46million can mean a lot...
especially with degrees of recursion
I had the exact same feeling
Just because thats the last they found, doesn't mean they haven't looked waaaaaaaaaaaaaaaaay bigger.
People, the joke is that there is a big difference between 46,000,000 and Nx10^46,000,000
“And it would be really weird if there was some massive massive gap and then some tri-perfect number”.
What about the sequence 1,3,tree(3)
I think you mean TREE(3), tree and TREE are different functions
@@waftingcloud it's a function involving coloured nodes that grows stupidly fast.
TREE(1)=1
TREE(2)=3
TREE(3)>Graham's number
The exact size of TREE(3) is not known, only lower limits and that it is finite.
Numberphile actually has videos on the TREE function that you can watch for an explanation of what the function does.
i like trees.
@@hewhomustnotbenamed5912 The exact size of TREE(3) is just one more than TREE(3) - 1
To be fair, TREE is based on a totally different problem, involving different objects than integers. And my (layman) understanding is that most functions in the Fast-Growing Hierarchy have boring values for x=1 and x=2, just due to the way they are constructed, so you always expect things to get interesting starting at 3. Tri-perfect numbers, on the other hand, grow at a regular pace for a while, with 6 values that steadily grow into the billions, then they just stop indefinitely.
"We think there are infinitely many of them but we've found fifty." I love maths so much XD
This is basically the same thing with mersenne primes though right
@@Catman_321 perfect numbers and mersenne primes are linked. Each even perfect number has a mersenne prime in its factors.
Dr James Grimes has a triperfect smile :)
:)
Bro Its cool to see you here
It's not him, his RUclips channel is singingbanana
false.
so perfect numbers are actually biperfect numbers
and the closest thing we have to perfect numbers are primes. They're themselves +1
or you could just say that 1 is the only perfect number...
CyanGaming | ᴹᶦᶰᵉᶜʳᵃᶠᵗ ⁻ ᴳᵃᵐᵉᴾᶫᵃʸ
Lim(P→+inf)(P+1) = P
tyab87 yet for all reals, p =/= p + 1
No, there's exactly one perfect number: The number 1. -1 also, if negatives are allowed.
-1 is a vector, not a number. 1 'pace' in the negative direction. So the number/quantity/magnitude is still 1 :)
Yes, this inconsistency annoyed me :)
Perfect numbers should be called biperfect numbers then.
Variety of Everything Or, better yet, tri-perfect numbers should be called bi-perfect numbers, so that way, we do not deal with the shenanigan that there would only be one mono-perfect number.
Or purrfect, purrrfect, purrrrfect, etc
@@matthijshebly 120 is a purrrfect number, but not a purrfect number.
Say that.
@@anawesomepet
This might lead to some confusion between rhotic and non-rhotic accents lol
@@anawesomepet Try to tell Isaac Arthur to say that.
I love these old school "pure" Numberphile videos where it really is just talking about some numbers. It's like fun mathematical trivia.
"As for how many we should sell in each pack, how do you like the number 6?"
"It's perfect"
Hexagons are the bestagons
@@Rudxainpacks of 7 cans?
That can save a lot of space during transport and storage...
@@maxaafbackname5562 6 around and 1 in the center? yeah!
The origin-story of 6-packs 👍🏻.
@@maxaafbackname5562 @Rudxain True 💡😃.
Fascinating! But I think there is a logic error around 5 minutes: "Either we'll find an odd perfect and therefore the list of triperfects is not complete, or the list is complete and there are no triperfects". There is still one remaining option on your truth table, there could be no odd perfects but still be more triperfects. Unless there is a proof you didn't mention that any new triperfects have to be related to an odd perfect.
Thank you! I was surprised no one else mentioned this.
Tulanir1 Because there was little need to mention it.
I love when James is featured in a video! He makes everything he explains sound twice as interesting
Minimum 11-perfect number seem to be is 2.518…E+1906. very very large!
Olbaid Fractalium that's gonna be my next favorite number
It'd be an interesting to define a sequence named R(n), such that R(n) is the smallest n-perfect number and then study this sequence.
@@angelmendez-rivera351 I was finding it interesting, that it seems like the number of n-perfect numbers is infinite when n=2, but then increases from six starting at n=3. I wonder why that is, and what it means about n=2, though I guess this is a number theorist's job lol
@@angelmendez-rivera351 it is also interesting if this sequence stop at some n. Or maybe just R(n) grows to infinitely big numbers.
@@OrangeC7 pretty sure they meant the smallest n perfect number. They said it in their comment.
For the record, the prime factorizations of the six triperefect numbers:
120 = (2^3) * 3 * 5
672 = (2^4) * 3 * 7
523,776 = (2^9) * 3 * 11 * 31
459,818,240 = (2^8) * 5 * 7 * 19 * 37 * 73
1,476,304,896 = (2^13) * 3 * 11 * 43 * 127
51,001,180,160 = (2^14) * 5 * 7 * 19 * 31 * 151
"The largest one we've found is the most recent one. It's 40 million.." -- "sounds big" -- "...digits long."
People wants to know why singing banana stopped singing.
Md. Sazzad Hossain want*
My bad😊
We think he stopped singing, and it would be weird if there was more after a gap this size, but nobody's been able to prove the set is finite.
Why did he stop making videos on his own channel? --that's what I meant.
@@iabervon lol
Are there n-perfect numbers for every n?
lol wt
And what about fractional-perfects, like 3/2 perfect?
well there is a 0-perfect number and a 1-perfect number.
I was hoping they'd finish the video with that point. Intuitively it feels to me like there shouldn't be, but the trend from 4-5-6-7 seemed to indicate there are more and more n-perfect numbers as n grows, but the numbers get larger and larger.
I think there are n-perfect numbers for every n as we can do the factors of n to find a n-perfect number based on m-perfect numbers with m
*_...not that it should be important but what-about 'sesquiperfect' (1.5×) and all the other 'rationally perfect' numbers..._*
2 is sesquiperfect
For any r = (p + 1)/p where p is prime, the only r-perfect number is p.
5:25 "I'm glad you asked"
This has to be the best moment in Numberphile history
after the parker square
Yesterday was perfect number day: 28/6. Also tau day, but I'm a pi bro.
15schaa 15schaa ew pi
I need to take note of this
Americans writing 6/28 are very sad, luckily I’m from Australia.
James needs to be on numberphile more often.
Yay James is my favourite person on this channel! He's always so full of energy!
4-perfect should be tetraperfect, and 5-perfect should be pentaperfect, and 6- hexa, and 7- hepta, etc.
How about 1/2 perfect numbers? (Where all the factors (except the last) add up to 1/2 the original number.) Perhaps call the semi-perfect numbers?
wouldn't that be the even numbers?
impossible, in order to half a number it must be even: so straight away, any possible candidate can be ruled since when halfing a number, it must also have factors 2 and every number has a factor of 1. so for example: 14: 7+2+1=10
Would there be another one besides 2?
2 is perhaps the only 'semi-perfect number'.
2 is the only one
I was hoping James would explain why if there is an odd perfect number then 2N would be triperfect
And 1-perfect numbers! (If you add up every factor of such a number, including itself, then you get itself).
The list of which is, in total: 1.
(0 is factored by everything, so it does not count).
We don't talk about 0.
0's factors are: 1,2,3,4,5,6,7... which add up to -1/12. Unless you don't count the number itself, in which case it's -1/12.
@@quinn7894 this is the best comment I have read in a while
3:21 - The Six Triperfect Numbers: 120 , 672 , 523776 , 459818240 , 1476304896 , 51001180160
Then would you call 1 a (the) monoperfect number?
CultistO Or uniperfect...
Yes.
Do you mean a number where the sum of his divisors equals itself?
Because if you do, the only "monoperfect" number is 1
Tuzeds Yes, exactly.
I'd enjoy a video about perfect numbers, triperfect numbers, etc in other bases besides base10.
wouldn't it be the same?
I suck at math, but still, I watch every video and try to understand. Maybe some day it all opens to me. Thank you for these videos.
For perfect numbers 4perfect, 5 perfect, to n-perfect what is the biggest “n” we have found?
That is just 2-perfect
:0
Uni-perfect numbers: where if you add up all the factors not including that number, you get 1.
I would really like to know the search algorithm for these perfect numbers. There certainly must be a clever twist to it, otherwise it would be impossible to check all numbers up to x* 10^30 and higher.
every even perfect number is represented in binary as p ONES followed by p − 1 ZEROS, e.g. 28 = 11100, 496 = 111110000, etc. Those are very easy to generate.
Raf M. That‘s pretty cool
Chr1z2 In fact, there is a specially fast algorithm used to find Mersenne primes (which is why all of the largest primes are Mersenne). Every Mersenne prime generates a perfect number, which is a more direct way of saying what a previous commentor was saying about binary.
I also know that Mersenne primes don't have to be checked by looking for divisors. There's a simpler test, that involves dividing the potential Mersenne prime into a number in a "square the last number and subtract 2" series. If it divides the corresponding number evenly, it's prime; if it doesn't, it's not.
The known triperfect numbers are 151-smooth (all the prime factors are
Let's say N is odd perfect.
Let f1..fn be its factors.
Then for each n, fn is odd and
2N=f1+f2+...+fn (perfect number)
We can show that there are exactly 2n factors of 2N, which are
F={f1, f2, ..., fn, 2f1, 2f2, ..., 2fn}
*[proof]*
*1. Members of F are factors of 2N*
Take k in 1..n, fk is odd and divides N therefore fk and 2fk both divide 2N.
*2. Factors of 2N are members of F*
Let f be a factor of N
Either f is even, f = 2g
2g divides 2N and 2 is prime to N therefore g divides N
Therefore g is in F and2g=f is in F
Or f is odd, f divides 2N and 2 is prime to N therefore f divides N
Therefore f is in F
*3. Members of F are distincts*
Take j ≠ k in 1..n, fj ≠ fk by definition therefore 2fj ≠ 2fk ; also fj ≠ 2fk since fj is odd ; fk ≠ 2fk since fk ≠ 0 by definition
*[/proof]*
Let's check if 2N is triperfect :
f1+...+fn+2f1+...+2fn
=f1+...+fn+2(f1+...+fn)
=3(f1+...+fn)
=3*2N
I love how their Gold and Silver RUclips Play Buttons are jsut casually sitting against the wall on the ground like that =]
As far as I know, 1 is the largest 1-perfect number.
Who instantly likes when they see Dr. James
Why are the picture frames not hanging on the wall?
Does 0 count? These are the factors of 0: 0. Let's add all those up: 0. 0 = 3 X 0. Mmmkay?
Literally every natural number is a factor of zero. And, as we all know the sum of the natural numbers is -1/12 (it really isn't, but who's counting)
bonecanoe86 Factors of 0: 0, 1, 2, 3... add them all up then.
0 is not a natural number so then it would be : 0: 1, 2, 3, ,4 ,5 , 6, 7.........Now add them all up .
And since -1/12 is ∞·0, zero is an ∞-perfect number.
If so, then zero is nth-perfect
Why aren't they called bi-perfect numbers? Cause a perfect number equals 1x its factors added up (minus the number itself).
And the described "triperfect" number equals 2x its factors added up (minus the number itself).
So it should be biperfect!
Couldn't help but notice that the larger the N for the n-perfect rule is, the more examples of it there are. Does that pattern hold up? And is there an obvious reason why that I didn't think of?
There’s a pretty obvious explanation for that. If you don’t know it then you should ask somebody who does, because I have no idea. I hate math.
That's not true..? There are 6 tri and 50 perfect
hey proto
The larger the number, the more factors there could potentially be.
Hmm
Do x-perfect numbers go off to infinity? like is there guaranteed to be one of at least 9-perfect, 10-perfect, 11-perfect.... numbers?
then couldn't you say 1 is the only monoperfect number, since it's the only number which adds up to itself if you add up all the factors (only 1).
Then all other primes would be near monoperfect; monoperfect + 1 in fact.
Kalkaanuslag Yes
Good old school numberphile.
i wonder if it's possible to extend this observation to all other numbers? For all possible ratios that you can achieve by dividing the sum of factors by the original number, for example a, are there only finitely many a-perfect numbers
How perfect are u sir?
One of the few things that still deeply fascinate and excite me is the fact that, despite the combined brain power of many very clever people over centuries, many very simple maths questions (such as: is there an odd perfect number?) cannot be definitively answered. Amazing.
Two questions:
How do they know that a hypothetical odd perfect number x2 would be a triple perfect number?
And what's the highest possible perfect number series? Can you have a 20-perfect number, a 100-perfect number? Where does it stop and why?
Conway79 The answer to your first question is simple. If N is odd perfect, the sum of its factors including N equals 2N. But when we introduce a factor of 2, we have double of each of these factors also on the list. Therefore, the sum of the factors of 2N is 2N+(2*2N)=6N meaning 2N is triperfect.
I suspect that the answer to your second question is not well-known.
Shouldn't perfect numbers be called "biperfect numbers" to fit the naming convention?
A James Grime video *breathing intensifies*
Two questions:
1. Is it true that James Grime is the brother of Frank Grimes, and only ditched the 's' so as to not arouse suspicion? and,
2. Like his brother, does he also like being called Grimey?
Now I'm curious about whether you get to a value of n for n-perfect numbers beyond which there are no numbers.
I can die, but cant miss any NUMBERPHILE video
Love for Maths
but with that naming convention, perfect numbers are actually diperfect numbers :(
that's annoyingly inconsistent
Tri perfect number = 3*sum
Di perfect/perfect number = 2*sum
So does that mean we also have monoperfect number =1*sum ... I mean 1 can be a monoperfect number ...
well, it totally is.
1½perfect number
2 is a 1½perfect number
1 is the only monoperfect. Primes have the lowest factors and they are all one too big
Dr. Grimmes, any Mersenne prime if is 2^k-1, then (2^k-1)(2^(k-1)) is a perfect number................... so.
Its Grime, horray!!!
So I suppose these n-perfect numbers are automatically antiprimes, aren't they?
Hi Dr Grime
Does anyone know why there are more 4-perfect numbers than tri-perfect numbers ?
46 million...
Wow that's a big num-
DIGITS long
Oh
What about true perfect numbers
All factors add to exactly the answer, the only example is 1
Do tri perfect number exists...??
But my mom said "no one is perfect"😭😭
Yeah but it only applies to human beings...
Not on numbers 😉😉😂
Not one number is perfect, however an infinitely many of them are
Numbers only need to fit a single definition of "perfect", while humans are faced with lots of different, and often conflicting, definitions of "perfect".
no one is perfect, but some tri are
That's because one isn't a perfect number.
I found something interesting while watching. Read my thoughts, what do you think?
TriPerfect numbers there are six. So 2 times 3.
6=2×3
4-perfect numbers there are 36.
36=4×9=2squared x 3squared
5-perfect numbers there are 65.
65=5x13=2+3×4+9 (the numbers used above)
I just think that is an interesting coincidence. Someone will tell me the obvious connection, but I don't know it better.
6-perfect numbers there are 245.
245=(2+3)×(4||9)
Can't find anything clever for 7's 516 so I think it's just a coincidence.
Hello, Numberphile. I know it's not in the subject, but I think I found a contradiction in your video: “ASTOUNDING: 1+2+3+4+5+…=-1/12” (Well, or I SSI). This video says that the sum of all natural numbers is -1/12, but it is also Alef0. So Alef0=-1 / 12. Alef0+1=Omega. Omega=11/12=121/144. Alef0*Alef0=Alef1. Alef1=1/144. Omega＞Alef1. Although, it may be that it is not, because the cardinals and ordinals are arranged in a certain order, but do not have a certain value. And sorry for my English, I'm just from Russia.
this is what happens when you think mathematics is about writing down meaningless symbols and shuffling them around. absolutely everything you wrote is nonsense
No, the sum of all natural numbers is not Aleph(0). There is absolutely no proof of this, nor did you provide a source.
Someone is sleeping with James. The police do not know who this is or what they are thinking.
If you start looking for N-perfect numbers, and you don't limit N to being an integer, how big can it get?
That doesn't make any sense, because that would turn any number into a N-perfect number. Take 42: it's a 16/7-perfect number.
Infinitely big
Well of course it makes sense. Maybe you find out, that the sum of all factors of a number is bounded by some constant, say sumoffactors(n)
Ah yes, the fabled sqrt(2)-perfect number... Yet to be found!
L4Vo5 N can be arbitrarily large. This is because the a abundancy index function is unbounded.
What about non-whole perfect numbers?
(Ex: 2 is a 1.5-perfect Number b/c 1+2 = 3 = 2 x 1.5)
every number is then in some way perfect, because the sum of its factors will always make up some fraction of the original
Matt B thanks. I would’ve also liked numberphile to show that in his video
James!
Woooooooooooooooosh
Woooooooooooooooosh
Woooooooooooooooosh
Woooooooooooooooosh
Woooooooooooooooosh
It's weird, right? How that dude is so FREAKING AWESOME in a totally nonsexual way?
Absolutley! How?!
He's pretty awesome in a sexual way too. 😉 Talk dirty to me math daddi
Non-sexual? Speak for yourself.
is he really unique in his nonsexualness combined with his freaking awesomeness to you?
I know plenty of people that I find freaking awesome in a nonsexual way, since for me awesomeness and sexualness aren't all that connected
false.
120 is the best number ever, no doubt. So many simple and elegant ways to write it :D
Perfect numbers are perfect because all their factors EXCEPT themselves add up to themselves.
So why are tri-perfect numbers not bi-perfect numbers? Why ignore the number itself for perfect numbers but not tri-perfect numbers? The only effect it would have would be shifting the number denoting the set down by one.
because when they started they wanted a cool name. And now 1 would be the only perfect number and that is a lonely number.
because normal perfect numbers aren't perfect, they're technically bi-perfect, but there are no real perfect numbers, because even a prime number n has a sum of factors of n+1, and it makes more sense this way than arbitrarily substracting the original number. perfect numbers are probably the first if this sort to be made up/ discovered, so they got the name.
Edit: as someone mentioned below, there actually is one perfect perfect number: 1.
Because mathematicians don't always name or define things in the most elegant way
Yeah it's kinda like how some genius decided that numbers off the number line should be called "imaginary", like as if numbers aren't imaginary already
Jason Martin Not only do they not always do it, they rarely do.
Here's my question: for all integers n, does there exist at least one n-perfect number?
The answer for all is 42
pft, that's not even a perfect number.
FabledDan I'm guessing you didn't get the reference. 42 is indeed the Answer to the Ultimate Question of Life, the Universe and Everything.
Indeed I have watched Hitchhiker's Guide to the Galaxy, it was just a joke :)
FabledDan ahhh... missed it through text. My apologies!
It was really 41 but people will forget you can start from 0 as well as 1. Display error.
I'd like a video about Marko Rodin's vortex mathematics please. I think that would be super fun.
Is there atleast 1 nth-perfect number for every n?
Zacharie Etienne That means the sum of all factors of the number, including the number itself, is the number itself.
Only one such number exists: 1
Abishek Anil you misunderstood the question
+Abishek Anil They mean “is there some n-perfect number for every value of n”, rather than “is n n-perfect”.
If you think about it 0 is a all n perfect solution. But I don't think that's a satisfying answer.
Shardar 1 no, zero doesn't work for a single finite n.... And I don't think the people calling it "infinity-perfect" are correct either, because technically each positive factor would have a negative counterpart. So it COULD satisfy the criteria for a "0-perfect" number but that's just my intuition, likely there is no 0-perfect number and 0 is not any type of perfect number
So.....perfect numbers are really biperfect. Or triperfect are really biperfect.
Either could be consistent in definition, but the actual definition isn't (which skips "bi" entirely and inexplicably).
Mephistahpheles Typical of mathematical nomenclature.
You're so
Precious
When you
*smile* ❤️
EdinPlayZ stfu
EdinPlayZ "oooooooooooollllllllllllldddddddd
Meme" is surely intellectual
Stelios Toulis Surely EdinPlayZ is a cancerous 9 year old
John Ox probably
EdinPlayZ i found it funny lol
Love the Mighty Nail and Gear in the background!
If numbers are infinite, why /wouldn’t/ there be more examples?
Well, not because the are infinite it means that there will be an infinite amount of them. A clear example is that there is one and only one number that is even and prime at the same time, and we know for sure that there isn't any other. Same thing may happen with these numbers.
If numbers are infinite, why is there only one example of even prime?
Konstanty because the only even prime is 2 and all other even numbers have at least one more factor than 1 and itself which is 2
Lol this is nonsense
No, this is Maths. Get used to it.
What I like about 6 is 1+2+3 = 6 and so does 1*2*3
The 6 perfect numbers would they be called... sexperfect?
Who's going to start a GoFundMe page to get James a hammer and some nails to finally hang those pictures?
Given that numbers go on infinitely, what if the list of perfect, tri-perfect, quad-perfect, etc. numbers actually does reappear after an incredibly long gap? Maybe they go thru this oscillating pattern where, from zero to infinity, there are clustered groupings of perfect numbers, followed by long gaps where there are none at all until one appears, initiating another cluster, then another incredibly long gap, and on and on and on...
seems likely...
we learnt this while discussing the domain and ranges of the function,
y=f(x)=(1/x) +1 and putting the ranges as domains and repeating it for certain iterations... numbers coalesce at a fixed pt 1.618... this happens and this can even happen for these perfect numbers.
Why are there more 4-perfect numbers than triperfect numbers?
So a perfect number should be called a biperfect number.
Thus a true perfect number is a number whose all factors add up to the number itself. There is only one true perfect number : 1.
Fun fact : The number of triperfect numbers is a biperfect number.
And I guess 0 is an infinityperfect number... ^^
Piffsnow 0 is what any set-theorist would call Aleph(0)-perfect.
We need squareperfect numbers. The sum of the factors of n is n^2. The one and only squareperfect number is 1, and then the list stops
I feel like they’d be biperfect numbers because you’re including the last factor which I don’t think you should do
Agreed, because that’s more consistent with the definitions of perfect and amicable numbers.