OEIS/3x+1 Levels

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Segments lengths

The length of a compressed segment is the number of nodes in its right part. The following lengths occur:

• 1 if the segment is constructed by a single µµ operation only, for left sides i ≡ 0, 2 mod 3 - a short segment. Such segments are targeted by rule 5 only.
• 3 if the segment is constructed by the 3 operation sequences µµ, µµδ, µµσ only, for left sides i ≡ 1 mod 3 - a medium segment. Such segments are targeted by rules 5 and 6 only.
• 5, 7 ... otherwise, also for left sides i ≡ 1 mod 3 - a variable segment. Such segments are targeted by rules >= 5. Rules >= 9 always target such a variable segment.

Medium and variable segments are longer than short ones.

Short segments attach to longer segments

In a first step of the attachment process we show that all short segments can be moved from subset E to D since they can be attached to a longer segment:

• Source segments with rules 6 and higher are attached to target segments 3*m + 1 which are longer by definition.
• Rule 5 attaches segments with LS = 4*k + 3 to target segments k + 1. We distinguish 3 cases:
  • For k = 3*m the target is 3*m + 1 and therefore longer.
  • For k = 3*m + 1 the source has LS = 4*(3*m + 1) + 3 = 12*m + 7, which is not short, so rule 5 never applies to these.
  • For k = 3*m + 2 the target is 3*(m + 1), to which we attach the source segment, and make it the new source 3*n. Then we look for the next target and find that either:
    • a rule >= 6 applies which leads to a longer segment, or
    • rule 5 is applicable for a source 3*n which also leads to a longer segment.

In total, E no longer contains short segments, since they were all disrooted and moved to D.

Medium segments attach to a segment in D or a variable one

We examine left sides i ≡ 1 mod 3, but we exclude segment (the root) for the moment. We are concerned with the cases where rules 5 or 6 are applicable, since otherwise the target is variable. As above, we start with rule 5:

i ≡ 1 mod 3 and i = 4*k + 3 and i > 1 => i ≡ 7 mod 12 = 7, 19, 31, 43 ... 
=> k = 1, 4, 7, 10 ... => targets 2, 5, 8, 11 ... => n ≡ 2 mod 3, 

Therefore these targets are short and already in D.

For rule 6 we have:

i ≡ 1 mod 3 and i = 4*k + 1 and i > 1 => i ≡ 1 mod 12 = 13, 25, 37, 49 ... 
=> k = 3, 6, 9, 12 ... => targets (3*k + 1) = 10, 19, 28, 37 ... => n ≡ 1 mod 9 and n > 1

I rule 5 applies for this target, then it is already in D. If some rule >= 9 applies, the claim of this section is also true. We are left with the cases where rule 6 is applicable again:

i ≡ 1 mod 9 and i = 4*k + 1 and i > 1 => i ≡ 1 mod 36 = 37, 73, 109, 145 ... 
=> k = 9, 18, 27, 36 ... => targets (3*k + 1) = 28, 55, 82, 109 ... => n ≡ 1 mod 27 and n > 1

Degrees of nodes

The degree of a node is the maximum number of possible nestings of the form 6*i - 2 in the number:

• 0 if the number is not of the form 6*i - 2 (white),
• 1 if the number is of the form 6*i - 2 (yellow),
• 2 if the number is of the form 6*(6*i - 2) - 2 (orange),
• 3 if the number is of the form 6*(6*(6*i - 2) - 2) - 2 (light red),
• 4 if the number is of the form 6*(6*(6*(6*i - 2) - 2) - 2) - 2 (dark red).

In the segment directories we use warm colors to indicate the degrees.

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