Is this a known pattern in mathematics? How would I represent it with an equation?

akunzler

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Nov 4, 2019
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Hey, thanks for the help!
I was messing around with math looking at the prime factors of a number as you manipulate it, and I came up with this pattern:
100 numbers:
1572923117978.png
1000 numbers:
1572923069819.png
...and the data has values for all positive integers.

It seems like a fairly obvious pattern: the y value increasing every 2^n numbers, (0 every 2, 1 every 4, 2 every 8, 3 every 16, etc.) and then repeating the previous numbers in a mirror-image sort of way before the y value gets higher 'the next time round'.
(So sorry if this is a bad explanation, but hopefully by looking at it you can see the pattern too)
Despite seeing the pattern, I am stumped in representing it as a function in terms of x. I see how I could get the values through a computer program using if-statements, but not mathematically.
Is this a known/named pattern? What function or equation could describe it?
Thanks a ton!
(If you want, here are the numbers: [0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,6,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,5,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,8,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,5,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,6,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,5,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,7,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,5,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,6,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,5,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,10,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,5,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,6,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,5,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,7,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,5,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,6,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,5,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,8,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,5,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,6,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,5,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,7,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,5,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,6,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,5,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,9,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,5,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,6,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,5,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,7,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1]
 

HallsofIvy

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You say "prime factors of a number as you manipulate it" but don't say how you are "manipulating" it. And your graph is x vs y without saying what x and y are. I assume that x is the given integer. Is y just the number of prime factors?
 

akunzler

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x is every (positive) integer and y is: how many factors of two are in (3*x + 1)... Yeah, kinda weird.
I'd like to find a function/equation to find any y value given x, but because it is a discrete function I don't know if that is possible. I am still very new in this area of math!
 

akunzler

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I'm continuing to experiment with it, and I feel like I am on to something. If I find the answer I will post it here.
 

akunzler

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Hmm...
You can represent it in terms of x with this, but I don't know how to put it in terms of y, especially how to work around Z.

Z being every (positive) integer, you can represent every point at y level by inputting any integer for \(\displaystyle y_{1}\) into the point(s)
\(\displaystyle \left(\frac{3\cdot2^{y_{1}+1}n_{all}+\left(-2\right)^{y_{1}}+\left(-1\right)^{2y_{1}+1}}{3},y_{1}\right)\)

Example graphed at \(\displaystyle y_{1} = 5\):
1573079846825.png
(black points are from data, blue points from the plotted point(s))
 

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lev888

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x is every (positive) integer and y is: how many factors of two are in (3*x + 1)... Yeah, kinda weird.
I'd like to find a function/equation to find any y value given x, but because it is a discrete function I don't know if that is possible. I am still very new in this area of math!
What does "how many factors of two are in (3*x + 1)" mean? I understand "how many factors (3*x+1) has". But "how many factors of two"?? Please give the first few values with explanation.
 

akunzler

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Sure!
I take the prime factors of 3x+1 (why x has to be an integer) and count how many (why y is always an integer) of them are 2.
So for 1, 3(1)+1 = 4 Prime factors of 4 are [2,2]. 2 of those are 2, so the y at 1 is 2.
For 2, 3(2)+1 = 7 Prime factors of 7 is [7] (it's a prime number). 0 of those are 2, so the y at 2 is 0.
For 3, 3(3)+1 = 10 Prime factors of 10 is [5,2]. 1 of those is 2, so the y at 3 is 1.
For 4, 3(4)+1 = 13 Also an odd number, no factors of 2. The y value at 4 is 0.
For 5, 3(5)+1 = 16 Prime factors of 16 is [2,2,2,2]. 4 of those are 2, so the y at 5 is 4.
For 6, 3(6)+1 = 19 Also an odd number, no factors of 2. The y value at 6 is 0.
etc.
I got all the values from a computer program that pretty much followed the above steps
 

lev888

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Got it.
The last answer on this page might be useful:
 

akunzler

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@lev888
It took me a little bit to understand that but I get it now, thanks for linking it.
Also btw I found out that I can plot the factors of two in a given x as a point at \(\displaystyle \left(x_{2},n\left\{\cos\left(\pi x_{2}2^{-n}\right)+1=0\right\}\right)\) or with \(\displaystyle \left(x_{2},\frac{\log\frac{\pi x_{2}}{2\pi n-\pi}}{\log(2)}\left\{\operatorname{mod}\left(\frac{\log\frac{\pi x_{2}}{2\pi n-\pi}}{\log(2)},1\right)=0\right\}\right)\) with \(\displaystyle x_{2}\) being an inputted x value and n being all integers. (though it really only needs to be integers up to ~1/2 x
And adapting that equation I made you can represent the above points with \(\displaystyle \left(x_{2},n\left\{\cos\left(\pi\left(3x_{2}+1\right)2^{-n}\right)+1=0\right\}\right)\)
 

Cubist

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Is this research related to the Collatz conjecture? If so then I wish you luck. It would be nice to see it proved/ disproved!
 

akunzler

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Is this research related to the Collatz conjecture? If so then I wish you luck. It would be nice to see it proved/ disproved!
It is, and thanks!
 

Cubist

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Your picture at the top very much reminded me of something yesterday. Now I have remembered.

In a search engine type "grey code" (or gray!) and switch to image results. You might have to scroll down a bit - but look for images where they have turned the code into a strip of black and white blocks to represent the 0 and 1. At first glance the dots in your pattern above match the transition points where black turns to white and vice-versa.

If I get time I'll have a go at reproducing the diagram. I didn't want to post an image link because it's against this site's policy.
 

akunzler

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Hmm, I can definitely see some similarities. I'll look into that.
Thanks!
 
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