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Prime numbers are often described with an example like the following: 'if you have n counters, and can't make a rectangle which has both sides longer than 1, n is prime'

I think it would be interesting to see what happens if we extend this to 'if you have n counters, you can't make a y dimensional 'cuboid' which has all sides longer than 1, n is prime(y)'

I.e. what we currently call prime numbers would be defined as prime(2)

and the first prime(3) numbers would be 2,3,4,5,6,7,9,10,11,13 etc.

Has anyone already looked at this? Or is there some trivial connection with 2-primes and y-primes that I'm missing?

JeffUK
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  • Your prime(3) numbers are those that have one or two (not necessarily distinct) prime factors. They are often called semiprimes The higher dimensions allow progressively more factors. – Ross Millikan Oct 13 '17 at 15:25
  • Apparently, repetitions are allowed, so the $y$-primes are the numbers $n$ with $\Omega(n)\le y$, where $\Omega(n)$ is the number of (not necessarily distinct) prime factors of $n$. – Peter Oct 13 '17 at 15:26
  • Your sequence gives 19 results in the OEIS. One of those might be relevant. – Robert Soupe Dec 12 '17 at 15:49

1 Answers1

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In number theory, the length of a natural number $n$ with a factorization like this: $$n=p_1^{r_1}\cdot\ldots\cdot p_m^{r_m}$$ is $r_1+\ldots+r_m$.

With your definition, a natural number is prime$(y)$ if its length is $\le y$.

ajotatxe
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