Part 2 - Gaps in Co-primes & some interesting properties

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First, let's discuss the following property:
For all N, the sequence if one of the following 2 types:
[Type 1] - It contains repeated instances of one distinct number. E.g. Sequences for Number[9], Number[25], Number[27], Number[49].
[Type 2] - If it contains more than one distinct number, then it contains all numbers from 1 to the largest number! (E.g. see Number[35] above, or Number [143] = -10-1-8-3-6-5-4-7-2-9-10-9-2-7-4-5-6-3-8-1-10- contains all numbers for 1 to 10).

For [Type 1], the reason is obvious. Take for example Number[27]. Since 27 is a power of 3, every 3rd number starting from 1 will share a factor 3 with 27. This leaves out the 2 numbers before every 3rd number, which are co-prime to 27, giving the sequence 2-2-2-2-2-2-2-2-2.

[Type 2] deserves more attention. "If the sequence contains more than one distinct number, then it contains all numbers from 1 to the largest number!" As examples, see pattern for
Number[35]    (4-1-2-3-4-3-2-1-4) ----> All numbers from 1 to 4,
Number[55]   (4-4-3-4-1-2-4-2-1-4-3-4-4),
Number[91]    (6-5-6-4-1-6-3-2-6-2-3-6-1-4-6-5-6) ----> All numbers from 1 to 6, or
Number[247]    (12-5-6-11-12-4-7-10-1-12-3-8-9-2-12-2-9-8-3-12-1-10-7-4-12-11-6-5-12) ----> All numbers from 1 to 12.

The sequence we are talking about is the clustering pattern of co-primes. The clusters are formed by appearances of non-coprime numbers that cause gaps in the co-prime cluster. It means that for any number N, the non-co-prime numbers appear at 'strategic' positions so as to form co-prime clusters of all possible lengths! Why in the world do they appear at such 'strategic' positions only? Why don't they want to miss any number (i.e. length) in between? This is really baffling. What is hidden in the nature of numbers or co-primes that causes this? As of now, I have no idea.

The next property is:

If k appears in N's sequence along with some other numbers less or more than k, then we can always find a number less than N whose sequence is made up entirely of k's.
Apart from examples mentioned on previous page, let's see another instance (for a larger number, just to make sure this property holds). Number [4489] consists of all 66's. This is the first time 66 appears in any sequence from Number[1] through Number[4489]. (Lets ignore the 'trivial appearance' for Number[67] for now). Later, 66 appears with other numbers when N is greater than 4489.

Why? Well, the reason just struck me while I was typing this up. Here's why (with an example) -
(1) For a number N, the largest element of the sequence is the (smallest prime factor of N) - 1. (This can be easily proved.)
(2) For multiples of 67, from 67 x 1 to 67 x 66, prime factors smaller than 67 prevent (67 - 1) from appearing in the sequence.
(3) The first time 66 gets a chance to appear in a sequence is when 67 is the smallest OR only the prime factor of N. The first such instance is N = 67 x 67. This allows 66 to appear in the sequence. This also being a square number, the pattern contains repeated instances of 66 alone.
(4) For multiples greater than 67, such as 67 x 68, etc., again prime factors smaller than 67 prevent 66 from appearing in the sequence. The first time 66 gets an opportunity to appear with other numbers is when 67 gets multiplied with a prime factor greater than 67, that happens to be 67 x 71.
(5) This is true for any prime factor p of number N (we can write a formal proof by replacing 67 and 66 in the explanation above with p and p-1).

The longer I look at these sequences, the more secrets it seems to reveal. For now, I guess I will pursue undertanding the "why" behind some the these properties. If you are reading this, thanks for hanging in there! I will be posting stuff as an when I discover more things about these sequences. Meanwhile, feel free to share your thoughts.

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