Over the last few months, I've been studying a pattern that I stumbled on concerning integers and their factors. First, I noticed that the number of factors a number has, follows an extremely regular pattern based on prime numbers. Meaning that starting with any prime and following multiples of that prime, the number of factors in those multiples will be the same. There are exceptions, but they are predictable. Each column below represents a prime number base and the number of factors for the prime multiple. It contains the primes from 2-199. I haven't seen this anywhere after a bit of searching.

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    $\begingroup$ It looks like you are just calculating $\frac{\sigma_{1}(N) - (N+1)}{N}$. For example, you can find the computation for $60$ (i.e., "$1.7833$") on WolframAlpha as follows: wolframalpha.com/input/?i=(sigma_1(60)-61)%2F60 $\endgroup$ – Benjamin Dickman Jul 2 '17 at 21:06
  • $\begingroup$ Yes, that is the basis for the calculation that generates the pattern represented in the images. Is this consistent pattern a known? $\endgroup$ – Troy W Jul 2 '17 at 21:55
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    $\begingroup$ "Thoughts?" It is not clear what pattern that "stems from every single number" you see in this. Open-ended discussions are not well suited for Math.SE. We seek high-quality content in the form of Questions (and Answers) that can be treated in a reasonably self-contained way. In its present form the Question is a call for open-ended discussion if not guesswork by Readers concerning what pattern interests you. $\endgroup$ – hardmath Jul 2 '17 at 23:10
  • $\begingroup$ "Every X numbers:" then followed by an image. Is it that you don't understand how the images are created? The image is created via the method I explained. I'm sorry you don't understand it but that shouldn't be misinterpreted as an "open ended" discussion. What question do you have about what I've shown? My question is to whether or not this pattern that stems from taking the sum of not-trivial divisors of N and dividing by N to produce this pattern is something that is known? It's a yes or no question. Explanations are great and appreciated. "Open ended" discussion is not what I'm after. $\endgroup$ – Troy W Jul 3 '17 at 0:54

Your 'less interesting' pattern has a very simple explanation.

The number of divisors $d(n)$ for a number $n = p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}$ is equal to $(e_1+1)(e_2+2)\cdots(e_k+1)$. This includes the $1$ and $n$ divisors, which is standard mathematical convention.

That means that any multiple $pk$ of a prime number $p$ will have $d(p)d(k) = 2d(k)$ divisors unless $p$ divides $k$, which is fairly rare for big $p$.

The sum of divisors of $n$ is $\displaystyle \sigma(n) = \prod_{i=1}^k \frac{p_i^{e_i + 1} - 1}{p_i - 1}$. Here as well we find that if $k$ and $p$ are coprime then the sum of divisors multiple $\sigma(kp) = \sigma(k)\sigma(p) = \sigma(k)(p+1)$. That's why for every row, when $\sigma(k)$ is big, the entire row is bright.

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  • $\begingroup$ Thanks, so is it also known that the ratios of this calculation at multiples of P will be consistent? I've seen lots of graphs on factors and numbers of factors but nothing like what I've come up with. Is there a reference to working with these particular calculations? If not, then it may be worth exploring more. If so, then I'm wondering what kind of work has been done. $\endgroup$ – Troy W Jul 2 '17 at 23:49
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    $\begingroup$ @TroyW It's a simple conclusion from the fact that the divisor function is multiplicative. $\endgroup$ – orlp Jul 2 '17 at 23:50
  • $\begingroup$ Simple as it may be, I'm wondering if it's characterized in published literature. The two are not mutually exclusive. $\endgroup$ – Troy W Jul 3 '17 at 0:09
  • $\begingroup$ @TroyW I think published literature would consider it trivial. I'm sorry :( I don't know exactly what you're looking for. $\endgroup$ – orlp Jul 3 '17 at 0:13
  • $\begingroup$ No problem, I hope to make it clear why I'm searching for published material. Thanks for your input! $\endgroup$ – Troy W Jul 3 '17 at 0:34

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