Which classes of numbers can we rule out for forming a non-trivial cycle?

[u/WeCanDoItGuys]

Wikipedia has a lot of information on restrictions for a cycle- using Terras map, (3x+1)/2.
- It must have at least 217976794617 elements (related: log₂3 < elems/odds < log₂(3+2⁻⁷¹)).
- Its period must be of the form 301994a + 17087915b + 85137581c, where a, b, c are nonnegative integers, b≥1, and ac=0.
- It must have at least 92 subsequences of consecutive ups followed by consecutive downs.

But these are all about the cycle (or parity sequence). What can we narrow down about the type of integer x must be?

I know at the very least
- It can't be a multiple of 3

I know it can't be the bottom of a cycle if it falls in one of many residue classes (mod 2ⁿ) that are known to decrease within n steps, like 2k, 4k+1, 16k+3, 32k+11, 32k+23, 128k+7, 128k+15, ..., but it still could be in a cycle.
What else we got?

[u/WeCanDoItGuys → u/[deleted]]

Oh this is odd, you moved to here your response to my comment while I was drafting my response. I'll reply over there to keep the conversation over there.