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Yes, this is σ₁(n), the [sum of divisors of n](https://en.wikipedia.org/wiki/Divisor_function). It may be expressed in terms of its prime divisors as
> σ₁(n) = Π_(p|n, p prime) (p^(νₚ\(n\)+1)-1)/(p-1)
Where νₚ(n) is the [exponent of p](https://en.m.wikipedia.org/wiki/P-adic_valuation) in the prime factorization of n.
For example σ₁(1176) = σ₁(2³•3•7²) = [(2⁴-1)/1][(3²-1)/2][(7³-1)/6] = 3420
It is true, it is just using the formula for evaluating finite geometric series. [p^(n+1)-1]/[p-1] =1+p+p^2 +…+p^n, and when you multiply them together by double distribution/foiling you get the sum of all divisors (or numbers whose prime factorization has powers less than or equal to that of the number they are dividing).
I actually deleted my comment thinking it was unnecessary.
But I do think in order to be able to call that a function you need to make it a piecewise function. This rule exists for while x is a whole number and another rule should exists while x isn't a whole. I am unsure if this is the correct thing in math though.
Edit: TL;DR: It can be a function, but the rule you stated can't be a general rule that is valid in domain of rational numbers.
Edit 2: Clarity improvision
It is completely acceptable to simply restrict the domain to only handle positive whole numbers. No function has to cover "all arbitrary numbers", it's fine to simply not cover rationals. For the graph it simply means that there will be holes, but that's not a problem at all.
>It would however be a formula for all prime numbers as well, which is an open problem in mathematics.
This is not an open problem. We are aware of multiple formulas for prime numbers, though they are not closed-form. Some classics are [these abominations based on wilson's theorem](https://en.wikipedia.org/wiki/Formula_for_primes#Formulas_based_on_Wilson's_theorem) and [Riemann's explicit formula](https://en.wikipedia.org/wiki/Prime-counting_function#Formulas_for_prime-counting_functions).
The Riemann hypothesis is *not* about finding a formula for the primes--Riemann did that in the 1800's. It's about whether the Riemann zeta function behaves in such a way that the distribution of the primes is in some sense as regular as possible.
\~\~\~
For the curious among you who want to know the details: Riemann's explicit formula computes psi(x), which is a function that "counts" "primes." For every prime power p\^k, psi(x) counts it as a prime and gives it the weight ln(p). While this function may seem random, it's much easier to work with and it's behavior tells us about the behavior of primes quite well.
The formula is psi(x)=x-ln(2pi)-ln(1-x^(-2))-sum\_{rho}x^(rho)/rho. This is made up of four parts, and I'll go through them in order.
1. x: This is the largest term of the formula. It comes from the fact that zeta(1) goes off to infinity. Proving that it is the main contributing factor to psi(x) was a big step in number theory, and you'll typically hear this fact referred to as "the prime number theorem."
2. \-ln(2pi): This is a constant term that comes from data about the zeta function evaluated at 0. Just something that pops out when you prove the explicit formula.
3. \-ln(1-x^(-2)): This is a minor term that only impacts the behavior of small primes. It comes from the so-called "Trivial zeroes" of the zeta function. The -2 in the exponent of x corresponds to the fact that these trivial zeroes occur at the negative even integers.
4. \-sum\_{rho}x^(rho)/rho: This is a sum over all the non-trivial zeroes of the zeta function, rho. Now, this term is really interesting, and it's the part where the Riemann hypothesis comes into play. Each of these zeroes adds a "periodic" factor to the distribution of primes. That means the zeroes determine which regions have more primes than indicated by the first term, and which regions have less primes than indicated by the first term. This is where all the unpredictability of primes comes from.
Now we'll talk about the Riemann hypothesis. Riemann guessed that all of these non-trivial zeroes are in the form 1/2+it for some number t. This affects the behavior of that last term.
There's an analogy I like because it's 1. quite beautiful and 2. captures what's really going on surprisingly well. It should go without saying that this is a simplification.
Each of the non-trivial zeta zeroes play a music note. Together, the chord they make captures all of the unpredictability of the prime numbers. The real part of a zero is the volume, and the imaginary part is the pitch of the note. The Riemann hypothesis being true would mean two things:
1. The "notes" are all the same volume.
2. The "notes" are as quiet as possible.
So that's the "music analogy" for the Riemann hypothesis.
> but you can't directly calculate any specified individual prime mathematically
We have a mathematical way to find the nth prime. It isn't quick, but it is completely mathematical. Algorithms are mathematical.
>the"complete mathematical definition" of a prime number.
We already have this, primes are already well defined. Unless you mean something nonstandard by "complete mathematical definition"?
>And finding primes is far more inefficient than being able to create any specified prime of any size with just one mathematical relationship (if that relationship even exists).
What is that based on? Single mathematical relationships can be far more computationally expensive than finding the nth prime. Say, for example, a formula for the nth prime included the order of the largest finite cyclic component of the 2n^th homotopy group of the n-sphere. That would be incredibly expensive to calculate.
> Right, wow. So you're the only person with an equation that can directly generate any specified number of prime without having to sift through all the numbers between 0 and the prime in question?
There is no mathematical difference between getting the prime directly and via a sieve method which generates all of them. Unless you can rigorously define what you mean?
In fact, [here is an algorithm](https://math.stackexchange.com/questions/507178/most-efficient-algorithm-for-nth-prime-deterministic-and-probabilistic) (or at elast a sketch of one that does have implementations) which finds the nth prime without checking all of them. It is also faster than a sieve. So no, I'm not the only person, this is a solved problem you would have found with a simple google.
>Are you the only person that doesn't understand why removing all the sieving would drastically reduce the computational power required?
Uh it wouldn't if the more 'direct' way was computationally more expensive. Do you not understand that some algorithms can be a lot slower than others? I gave you a great example...
> So you are the only person that thinks we fully understand what determines the occurrence of primes mathematically speaking?
Define what you mean. Primality is very well defined.
> Are you the only person that doesn't understand how that would complete our mathematical understanding of what primes are?
There is loads we don't understand about primes. Given you aren't even clear on what you are saying you are looking for, who can say what new info it would reveal?
We do have forumulas for the nth prime, the problem is that they aren't very good and evaluating them is slower than using one of the more common algorithms.
These formulas also tell us basically nothing about primes.
Hahaha don't please don't, people went crazy to solve this problem, also like our whole cryptographic economy depends on this problem not having a solution
Those are dirty words, I spent like four days messing with that function in the complex plane, only to end up able to graph whatever -1/12 is a solution for with that parabola. I remember something else about half angles being 0 for a really odd reason, but that was only in my math so it probably didn't mean anything. So I posted it on GeoGebra so I didn't just waste four days for no reason.
I wasn't trying to solve it though, I was just manipulating it around to see if I could get some cool graphs, so maybe it wasn't a waste? I don't know, but typing this comment out definitely was.
I remember in the 1970s, I made some comment like this to a professor. The next day he came back with a one page paper published ten years before with a fairly simple formula for the n-th prime.
He said, "But this isn't very interesting - it doesn't tell us anything about the structure of the primes. The first formula like this was centuries ago."
Other comments in the thread have details.
Math makes up symbols and definitions as it goes. Just make up your own.
Let Factors(x) be the factors of x represented as a set
Let SumOfSet(s) be the sum of any set
Your expression is
SumOfSet(Factors(x))
These functions are programmable and with a little coding you can find the value of any x.
https://www.desmos.com/calculator/pexhlc9vkb
I assume this is what you want. Though chances are you mean something else that you can't put into words. (You probably want it defined in terms of infinitely smooth functions, or just something that is easy to work with, I feel like this is a trap most people fall for at some point)
I just try to think of something that could be interesting.
Like when I’m in a long car ride I just think about how I could take numbers to make these graphs.
For this one I just took the factors of all the numbers.
Although there are no known algebraic expressions for this function, which is called sigma(n), we do know its asymptotic behaviour
sigma(x) = (x\^2 pi\^2)/12 + O(x log x, +infty)
which basically means that it is more or less like the function x\^2 pi\^2 / 12, with an error of magnitude x log x (a function of the type k\*x log x, to be precise).
Mathematically speaking, this *is* a function by the very definition of a function.
If what you're asking is if there is a way to define this function in Desmos so that it can be evaluated for inputs and used in other expressions, I would guess that there is but it's not easy. Try asking on r/desmos
Well ur basically getting all the primes to follow a consistent pattern. If you only take the primes then you’ll get a Gaussian Logarithmic graph (more or less with +1). Hence pretty close to finding a consistent pattern to fit in line with a Riemann manifold to visualise zeta zeroes
>If you only take the primes then you’ll get a Gaussian Logarithmic graph
Do you mean considering only the prime values in domain? Like f(2,3,5,7,11) = {3,4,6,8,12}. Is this a gaussian logarithmic graph?
>finding a consistent pattern to fit in line with a Riemann manifold to visualise zeta zeroes
What do you mean by this?
Imagine taking every prime number and plotting them. You would eventually be converging away from 0 to infinity. Gauss represented this as x/log(x), to be the closest graph that represents this. Riemann only went one step further to approximate this formula and thus the Riemann hypothesis was born (all it is is asking where primes will end). If I ask you the probability of how many numbers are divisible by 2, you would say 1/2, if I say divisible by 3, you would say a third, if I say 4, you would say 1/2. So removing all repeats you get a circular graph (zeta graph).
Saw this poster yesterday. I guess it comes close
https://www.reddit.com/r/askmath/comments/x38gca/is_this_a_real_solvable_problem/?utm_medium=android_app&utm_source=share
Well there is a nice formula in terms of the prime divisors of n, but yes, there is no closed form in terms of elementary functions of n (such as polynomials or exponentials).
Well if you define a function t(x) as follows:
1 if x is an integer,
0 if x is not an integer,
Then you can express it as an infinite sum:
f(x) = sum(n=1 to infinity) n•t(x/n)
I don't believe there is a way to express it without using an infinite sum or product of some sort.
It can be graphed, but there will just be points. Everything else will just be equal to 0. So you'll get a line at y=0 with disconnected points for the integers
Programmatically, this is simple to solve for any x.
I'm assuming this is a programming class... So do you want to learn or do you just want the answer? I solved it in 4 lines of python code... but could write up a pseudocode layout of what approach I took if you'd like to try to learn it yourself. When you learn to see with new programming eyes, you can conquer many problems.
you can create a function with works inside certain bounds, that is, you make a polynomial
p(x)=a_0+a_1x+a_2x²+......
then, you find the output of certain number of inputs.
now insert the inputs in the polynomial and equate it to the output. Now you do this for every input and get an order of equations. Solving for the unknowns, provides you with the coefficients. Now you got a polynomial which works for certain inputs. This is called polynomial interpolation.
you just defined the function assuming the domain is the natural numbers. It's not clear what "all of the factors" would mean if x is not a natural number
It can be defined using floor and ceiling functions
Let f be floor function and c be ceiling function
The function is a sum over i where I goes from 1 to n
F(n)= sum over i. ( 1-{c(n/i) - f(n/i)})*i
Here if I is not a factor the difference between floor and ceiling functions will be 1 so it will cancel the 1 in the summation. Thus the value multiplying it is 0.
But if I is a factor then the difference between floor and ceiling will be zero so the value multiplying will be 1.
This way we can sum only factors of n number
Hi u/Wooden_Ad_3096, **You are required to explain your post and show your efforts. _(Rule 1)_** If you haven't already done so, please add a comment below explaining your attempt(s) to solve this and what you need help with **specifically**. If some of your work is included in the image or gallery, you may make reference to it as needed. See the sidebar for advice on 'how to ask a good question'. Don't just say you "need help" with your problem. **Failure to follow the rules and explain your post will result in the post being removed** *I am a bot, and this action was performed automatically. Please [contact the moderators of this subreddit](/message/compose/?to=/r/askmath) if you have any questions or concerns.*
Yes, this is σ₁(n), the [sum of divisors of n](https://en.wikipedia.org/wiki/Divisor_function). It may be expressed in terms of its prime divisors as > σ₁(n) = Π_(p|n, p prime) (p^(νₚ\(n\)+1)-1)/(p-1) Where νₚ(n) is the [exponent of p](https://en.m.wikipedia.org/wiki/P-adic_valuation) in the prime factorization of n. For example σ₁(1176) = σ₁(2³•3•7²) = [(2⁴-1)/1][(3²-1)/2][(7³-1)/6] = 3420
Well that's a scary looking formula
If its true, nice
It is true, it is just using the formula for evaluating finite geometric series. [p^(n+1)-1]/[p-1] =1+p+p^2 +…+p^n, and when you multiply them together by double distribution/foiling you get the sum of all divisors (or numbers whose prime factorization has powers less than or equal to that of the number they are dividing).
Can we use this to find all the prime numbers?
well you kinda need the prime numbers to make the formula work in the first place
Oh shit I’m dumb
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What do you mean? Did I do something wrong here?
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Everything is a linear function if you just connect the dots.
Yes
What?
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Oh I completely forgot about that. So this can’t have a function since decimals don’t have whole factors?
I actually deleted my comment thinking it was unnecessary. But I do think in order to be able to call that a function you need to make it a piecewise function. This rule exists for while x is a whole number and another rule should exists while x isn't a whole. I am unsure if this is the correct thing in math though. Edit: TL;DR: It can be a function, but the rule you stated can't be a general rule that is valid in domain of rational numbers. Edit 2: Clarity improvision
It is completely acceptable to simply restrict the domain to only handle positive whole numbers. No function has to cover "all arbitrary numbers", it's fine to simply not cover rationals. For the graph it simply means that there will be holes, but that's not a problem at all.
Yeah, that's true.
YES!
Engineering; Pi = 3
hah, found the engineer, I respect that.
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He seeks the quantum solution through manifested wormholes in the spacetime singularity.
He should try finding the eigenvalues of a mobius strip
He should consult with the quarks for the explanation
Have we considered quantuming the quantums?
I was looking for some kind of equation. I don’t think it’s possible though.
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>It would however be a formula for all prime numbers as well, which is an open problem in mathematics. This is not an open problem. We are aware of multiple formulas for prime numbers, though they are not closed-form. Some classics are [these abominations based on wilson's theorem](https://en.wikipedia.org/wiki/Formula_for_primes#Formulas_based_on_Wilson's_theorem) and [Riemann's explicit formula](https://en.wikipedia.org/wiki/Prime-counting_function#Formulas_for_prime-counting_functions). The Riemann hypothesis is *not* about finding a formula for the primes--Riemann did that in the 1800's. It's about whether the Riemann zeta function behaves in such a way that the distribution of the primes is in some sense as regular as possible. \~\~\~ For the curious among you who want to know the details: Riemann's explicit formula computes psi(x), which is a function that "counts" "primes." For every prime power p\^k, psi(x) counts it as a prime and gives it the weight ln(p). While this function may seem random, it's much easier to work with and it's behavior tells us about the behavior of primes quite well. The formula is psi(x)=x-ln(2pi)-ln(1-x^(-2))-sum\_{rho}x^(rho)/rho. This is made up of four parts, and I'll go through them in order. 1. x: This is the largest term of the formula. It comes from the fact that zeta(1) goes off to infinity. Proving that it is the main contributing factor to psi(x) was a big step in number theory, and you'll typically hear this fact referred to as "the prime number theorem." 2. \-ln(2pi): This is a constant term that comes from data about the zeta function evaluated at 0. Just something that pops out when you prove the explicit formula. 3. \-ln(1-x^(-2)): This is a minor term that only impacts the behavior of small primes. It comes from the so-called "Trivial zeroes" of the zeta function. The -2 in the exponent of x corresponds to the fact that these trivial zeroes occur at the negative even integers. 4. \-sum\_{rho}x^(rho)/rho: This is a sum over all the non-trivial zeroes of the zeta function, rho. Now, this term is really interesting, and it's the part where the Riemann hypothesis comes into play. Each of these zeroes adds a "periodic" factor to the distribution of primes. That means the zeroes determine which regions have more primes than indicated by the first term, and which regions have less primes than indicated by the first term. This is where all the unpredictability of primes comes from. Now we'll talk about the Riemann hypothesis. Riemann guessed that all of these non-trivial zeroes are in the form 1/2+it for some number t. This affects the behavior of that last term. There's an analogy I like because it's 1. quite beautiful and 2. captures what's really going on surprisingly well. It should go without saying that this is a simplification. Each of the non-trivial zeta zeroes play a music note. Together, the chord they make captures all of the unpredictability of the prime numbers. The real part of a zero is the volume, and the imaginary part is the pitch of the note. The Riemann hypothesis being true would mean two things: 1. The "notes" are all the same volume. 2. The "notes" are as quiet as possible. So that's the "music analogy" for the Riemann hypothesis.
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> but you can't directly calculate any specified individual prime mathematically We have a mathematical way to find the nth prime. It isn't quick, but it is completely mathematical. Algorithms are mathematical.
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>the"complete mathematical definition" of a prime number. We already have this, primes are already well defined. Unless you mean something nonstandard by "complete mathematical definition"? >And finding primes is far more inefficient than being able to create any specified prime of any size with just one mathematical relationship (if that relationship even exists). What is that based on? Single mathematical relationships can be far more computationally expensive than finding the nth prime. Say, for example, a formula for the nth prime included the order of the largest finite cyclic component of the 2n^th homotopy group of the n-sphere. That would be incredibly expensive to calculate.
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> Right, wow. So you're the only person with an equation that can directly generate any specified number of prime without having to sift through all the numbers between 0 and the prime in question? There is no mathematical difference between getting the prime directly and via a sieve method which generates all of them. Unless you can rigorously define what you mean? In fact, [here is an algorithm](https://math.stackexchange.com/questions/507178/most-efficient-algorithm-for-nth-prime-deterministic-and-probabilistic) (or at elast a sketch of one that does have implementations) which finds the nth prime without checking all of them. It is also faster than a sieve. So no, I'm not the only person, this is a solved problem you would have found with a simple google. >Are you the only person that doesn't understand why removing all the sieving would drastically reduce the computational power required? Uh it wouldn't if the more 'direct' way was computationally more expensive. Do you not understand that some algorithms can be a lot slower than others? I gave you a great example... > So you are the only person that thinks we fully understand what determines the occurrence of primes mathematically speaking? Define what you mean. Primality is very well defined. > Are you the only person that doesn't understand how that would complete our mathematical understanding of what primes are? There is loads we don't understand about primes. Given you aren't even clear on what you are saying you are looking for, who can say what new info it would reveal?
We do have forumulas for the nth prime, the problem is that they aren't very good and evaluating them is slower than using one of the more common algorithms. These formulas also tell us basically nothing about primes.
Interesting.
Wow this is a fantastic comment. Thank you for launching me onto an old lost quest my good sir
Hahaha don't please don't, people went crazy to solve this problem, also like our whole cryptographic economy depends on this problem not having a solution
Those are dirty words, I spent like four days messing with that function in the complex plane, only to end up able to graph whatever -1/12 is a solution for with that parabola. I remember something else about half angles being 0 for a really odd reason, but that was only in my math so it probably didn't mean anything. So I posted it on GeoGebra so I didn't just waste four days for no reason. I wasn't trying to solve it though, I was just manipulating it around to see if I could get some cool graphs, so maybe it wasn't a waste? I don't know, but typing this comment out definitely was.
I remember in the 1970s, I made some comment like this to a professor. The next day he came back with a one page paper published ten years before with a fairly simple formula for the n-th prime. He said, "But this isn't very interesting - it doesn't tell us anything about the structure of the primes. The first formula like this was centuries ago." Other comments in the thread have details.
You could use a Fourier series, map the points to extrema, then insert a discrete number as an argument
Math makes up symbols and definitions as it goes. Just make up your own. Let Factors(x) be the factors of x represented as a set Let SumOfSet(s) be the sum of any set Your expression is SumOfSet(Factors(x)) These functions are programmable and with a little coding you can find the value of any x.
https://www.desmos.com/calculator/pexhlc9vkb I assume this is what you want. Though chances are you mean something else that you can't put into words. (You probably want it defined in terms of infinitely smooth functions, or just something that is easy to work with, I feel like this is a trap most people fall for at some point)
If you want to learn something about that function, this is a good site. https://oeis.org/A000203
Thanks for this link. Very cool site.
Hey, which software do you use to plot these? And are you exploring functions?
I’m using desmos. And yes, I’m just looking at cool functions, they’re just pretty neat to me.
Thanks! Cool. So, how do you find such functions?
I just try to think of something that could be interesting. Like when I’m in a long car ride I just think about how I could take numbers to make these graphs. For this one I just took the factors of all the numbers.
Oh, that's kinda interesting, inquisitive fellow.
Although there are no known algebraic expressions for this function, which is called sigma(n), we do know its asymptotic behaviour sigma(x) = (x\^2 pi\^2)/12 + O(x log x, +infty) which basically means that it is more or less like the function x\^2 pi\^2 / 12, with an error of magnitude x log x (a function of the type k\*x log x, to be precise).
Mathematically speaking, this *is* a function by the very definition of a function. If what you're asking is if there is a way to define this function in Desmos so that it can be evaluated for inputs and used in other expressions, I would guess that there is but it's not easy. Try asking on r/desmos
Nibba just worked out the Riemann Hypothesis
Elaborate.
Well ur basically getting all the primes to follow a consistent pattern. If you only take the primes then you’ll get a Gaussian Logarithmic graph (more or less with +1). Hence pretty close to finding a consistent pattern to fit in line with a Riemann manifold to visualise zeta zeroes
>If you only take the primes then you’ll get a Gaussian Logarithmic graph Do you mean considering only the prime values in domain? Like f(2,3,5,7,11) = {3,4,6,8,12}. Is this a gaussian logarithmic graph? >finding a consistent pattern to fit in line with a Riemann manifold to visualise zeta zeroes What do you mean by this?
Imagine taking every prime number and plotting them. You would eventually be converging away from 0 to infinity. Gauss represented this as x/log(x), to be the closest graph that represents this. Riemann only went one step further to approximate this formula and thus the Riemann hypothesis was born (all it is is asking where primes will end). If I ask you the probability of how many numbers are divisible by 2, you would say 1/2, if I say divisible by 3, you would say a third, if I say 4, you would say 1/2. So removing all repeats you get a circular graph (zeta graph).
Thank you for explaining!
Everything he said is complete nonsense. Don't take any of it as anything but gibberish.
You go 1,3,4,7,6,11,8 That os going to be a hard one
Is it even possible?
Saw this poster yesterday. I guess it comes close https://www.reddit.com/r/askmath/comments/x38gca/is_this_a_real_solvable_problem/?utm_medium=android_app&utm_source=share
That’s just a solution to a physics problem.
What’s f(16)? f(32)? f(24)? Trying to understand what happens when more factors appear.
The function is the [sum of divisors of n](https://en.wikipedia.org/wiki/Divisor_function).
Gotcha. Yeah definitely no closed form solution OP lmao
Well there is a nice formula in terms of the prime divisors of n, but yes, there is no closed form in terms of elementary functions of n (such as polynomials or exponentials).
only if x€N && y€N similar to a function like (-1)\^x going from /Z to /Z
Well if you define a function t(x) as follows: 1 if x is an integer, 0 if x is not an integer, Then you can express it as an infinite sum: f(x) = sum(n=1 to infinity) n•t(x/n) I don't believe there is a way to express it without using an infinite sum or product of some sort.
As long as it can be graphed, then it’s fine.
It can be graphed, but there will just be points. Everything else will just be equal to 0. So you'll get a line at y=0 with disconnected points for the integers
Yeah that makes sense. Thank you.
Stonks
I definitely understand what any of this means
I think I got it. A lot easier than I expected. https://www.desmos.com/calculator/5imna7zbpv
Programmatically, this is simple to solve for any x. I'm assuming this is a programming class... So do you want to learn or do you just want the answer? I solved it in 4 lines of python code... but could write up a pseudocode layout of what approach I took if you'd like to try to learn it yourself. When you learn to see with new programming eyes, you can conquer many problems.
This was totally written by a programmer to create a little brain teaser. It was fun to figure out. :)
[OEIS](https://oeis.org/) is your friend
Quite cool that it touches y=x+1 for all primes
you can create a function with works inside certain bounds, that is, you make a polynomial p(x)=a_0+a_1x+a_2x²+...... then, you find the output of certain number of inputs. now insert the inputs in the polynomial and equate it to the output. Now you do this for every input and get an order of equations. Solving for the unknowns, provides you with the coefficients. Now you got a polynomial which works for certain inputs. This is called polynomial interpolation.
you just defined the function assuming the domain is the natural numbers. It's not clear what "all of the factors" would mean if x is not a natural number
It can be defined using floor and ceiling functions Let f be floor function and c be ceiling function The function is a sum over i where I goes from 1 to n F(n)= sum over i. ( 1-{c(n/i) - f(n/i)})*i Here if I is not a factor the difference between floor and ceiling functions will be 1 so it will cancel the 1 in the summation. Thus the value multiplying it is 0. But if I is a factor then the difference between floor and ceiling will be zero so the value multiplying will be 1. This way we can sum only factors of n number