# Guess the Function for Scatterplot of Number Theoretic Function

To my knowledge, this puzzle is not previously published (except by me on google+ recently), but I would be interested to hear of any info otherwise.

This following graph was generated using a simple ruby program and gnuplot. It graphs a "basic/fundamental" number theoretic property of the natural numbers. Can you figure out what it is? Increasingly detailed hints below if you want more help. The answer and further background will be posted as a comment after some time.

hint1:

"fundamental" as in "fundamental theorem of arithmetic"

hint2:

one word: prime

hint3:

prime decomposition

• What does mean "graphs a "basic fundamental" number theoretic property of the natural numbers"? Can you give an example how you graph a property of numbers? – klm123 Aug 8 '14 at 20:30
• let x be a natural number and y=f(x) be some simple number-theoretic function of x. – vzn Aug 8 '14 at 20:37
• and what then? How do you graph it? take all possible natural numbers X and put (y,x) points on the graph? – klm123 Aug 8 '14 at 20:50
• as axis labels on graph state x ∈ [1..1000] – vzn Aug 8 '14 at 21:09

$x$ against its largest prime factor. The points on the top line are primes, the next line down are primes times two, then primes times three, and so on, with the scattering at the bottom being as a result of numbers that are a product of more than two primes.