I was playing around with the Collatz sequences of numbers up to a number. My question was, for which numbers $n$ does no integer below $n$ reach $n$ in its iteration? So I wrote a program to find all the ones that have that property and I quickly realized all multiples of 3 do. Obviously, because $3n + 1$ can never be a multiple of 3 and if it's not already a multiple of 3, dividing by 2 obviously won't make it one.

However, what I noticed after that is that the rest of the numbers are almost all primes. Only some primes, however, and I'm not sure what the criteria for it is. A small percent of them are also numbers that have some number of prime factors, and as you go up into the higher numbers of course the number of prime factors on average increases. None of these numbers are even for unknown reasons to me that I'm sure I could figure out if I looked hard enough.

Consequently, the percent of these numbers that are prime decreases as you go for higher values of $n$, but remains higher than you'd expect. Of course, there are lots of primes because any number in the list has to be odd, and can't be divisible by 3 because I excluded those because they're trivial.

However, my question is, what, if any, are the criteria for any number that satisfies this property (no integer below $n$ reaches $n$ in its iteration)? Besides not being a multiple of 3 I couldn't come up with any solid reasoning.

As a bonus, is it really true that all numbers even and not divisible by 3 have some number below it whose sequence includes it? And if so, why?

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    $\begingroup$ For numbers up to 100000: 43266 have this property, 29763 of those are multiples of 3, 7964 are two primes multiplied, 2851 are primes, 2687 are three primes multiplied or higher $\endgroup$ – TigerGold Nov 15 '19 at 3:39
  • $\begingroup$ I would type the numbers into the Online Encyclopedia of Integer Sequences to see what turns up. $\endgroup$ – Gerry Myerson Nov 15 '19 at 4:56
  • $\begingroup$ $2x+1$ goes through $6x+4$ and $3x+2$ thus elimination all primes that have remainder 5 on division by 6. $\endgroup$ – user645636 Nov 15 '19 at 12:11
  • $\begingroup$ where does $2x+1$ come into it $\endgroup$ – TigerGold Nov 15 '19 at 20:54
  • $\begingroup$ So have you consulted the OEIS yet? $\endgroup$ – Gerry Myerson Nov 16 '19 at 11:27

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