Prime factors

Like many people, I am obsessed with prime numbers. To understand more about them, I plotted a 2D diagram of the natural numbers, showing for each the prime factors and where they occur. I was astonished to find obvious regularities that, in hindsight, make sense, but that I had never encountered before.

Here's the diagram, made manually on a spreadsheet. I'd love to hear your observations.

Here are mine:
* Every number repeats as a factor at its own frequency: 2 repeats at every 2 numbers, 3 at every numbers, etc.
* The numbers where no previous frequency repeats are prime numbers.
* Each number starts its own sequence, displayed here as a line with different slope and color. What is the significance of this slope?
* In this 2D space, one octant is enough to represent all numbers and all factorizations.

AttachmentSize
numbers.png58.69 KB
numbers.ods17.83 KB

101?

come on, this should be at least 201 ;)

The slope is the number

The slope is the number itself * -1.

So the lines are: -2*x-1 for the number 2 -3*x-2 for the number 3 -4*x-3 for the number 4 and so on...

Right! the slope is the

Right! the slope is the number, the frequency is also the number. I wonder if this representation can help find a closed form for the prime sequence.