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Say there was a function f(x) which would display integers based on how composite they were, and numbers that had more factors than others would show up as spikes on the graph.

Take the number 840. This number has 32 distinct factors, and according to a program I ran, this is the most out of any number in the range 1-1,000. Because of this, it would show up as a big spike on the graph. Now, take the number 3,510. This number also has 32 distinct factors, but because it is much larger, there are many more numbers in its range which have more distinct factors, and as such, it will not be as big of a spike as 840.

Now, take the number 397. It is a prime number, and a very large one. Because of this, it would be a very low point on the graph. But, take the number 5. It is also prime, but being such a low prime, there are not very many numbers in its range that have more factors. So, it would be a lot higher than 397 on the graph.

Is there a function that can do anything like this?

eners49
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    The divisor function, $\sigma_0(n)$ just counts the number of divisors of $n$, for example. – lulu Feb 07 '21 at 22:59
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    So you want numbers with lots of divisors to stand out but you also want smaller numbers with lots of divisors to stand out more than larger numbers with the same numbers of divisors. So how about $d(n)/n$, where $d(n)$ (a.k.a., $\sigma_0(n)$) is the number of divisors of $n$? I think you are going to need to experiment until you find a function whose graph you like. E.g., $d(n)/\log(n)$ might look better than $d(n)/n$. – Rob Arthan Feb 07 '21 at 23:05
  • Is there a good approximation for the divisor function? For example, the prime-counting function is approximated as x/(ln x). What about the divisor function? – eners49 Feb 07 '21 at 23:17
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    I don't see how there can be a good closed form approximation for the divisor function: as you observe it is very spiky. Are you concerned about efficient ways to calculate it? – Rob Arthan Feb 07 '21 at 23:48
  • There is an approximation for the divisor summatory function (see https://en.wikipedia.org/wiki/Divisor_summatory_function#Dirichlet's_divisor_problem). – Steven Clark Feb 08 '21 at 03:22
  • A couple of related functions include the prime omega functions (see https://en.wikipedia.org/wiki/Prime_omega_function) which count the number of primes dividing n with and without multiplicity. – Steven Clark Feb 08 '21 at 03:29
  • What about $\Omega(n)$ (the number of prime factors , possibly repeated) ? The larger $\Omega(n)$ , "the more composite" is $n$. – Peter Feb 09 '21 at 11:19
  • For some huge $n$ , we won't be able to determine the number of divisors in practice since determine the number of divisors requires prime factorization. – Peter Feb 09 '21 at 11:22
  • What might interest you are the "highly composite numbers" – Peter Feb 09 '21 at 11:23

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