Comment to the program 'Sequence of least odds of Collatz classes'

The program
The program calculates the least odds of Collatz classes (see Definitions below), having 12 digits at most. The corresponding Collatz divisors (see Definitions) are calculated up to 237 inclusive.
When ordered by size, these odds are the first terms of an infinite sequence. The program has the possibility to order the sequence by the size of the Collatz divisor of each term, which makes it easier to find some important (infinite) subsequences (to be discussed in Sequences down here). The program is downloadable (click button 'To Free Download Page').

Background
The program is a tool for investigation of Collatz sequences defined from odd to odd (see Definitions). In the notation of Collatz sequences going from odd to odd, the Collatz divisor d = 2x is shown between successive odds (between parentheses). Formally and in general this is not a correct notation for sequences, however, it is easy to 'translate' the sequence back to 'formal' and it is to great advantage of the investigation of relations of odds such as in Collatz classes, composite Collatz classes and more. The sequences presented here have the common (formal) notation. However, they deal with Collatz classes and Collatz divisors, which are ideas strongly connected to the above notation of Collatz sequences going from odd to odd.

Calculation of least odds of Collatz classes
A Collatz class C(m,d) = {p | p = am,d + h * 6 * d} in which am,d is the least odd of the set and h ∈ {0, 1, 2, ...}. All odds p of the set have p ≡ m (mod 3) and their Collatz divisor = d = 2x. When the least odd am,d of the set is known, all p of the set can be calculated. The iterative function the program uses to calculate am,d for all pairs (m,d) if m = 0, 1, 2 and d = 2, 4, 8, 16, 32, 64 ,... has already been discussed in the Comments to some other of our programs (see Other relevant programs on this site, below). We repeat some aspects of it in Discussion of the method of calculation, down here.

Define an iterative function G(p0) = 4p0 + 1 | p0 ∈ N+ ∧ p0 is odd, then p0, G1(p0), G2(p0), in general Gk(p0) have equal Collatz iterates p1. The corresponding Collatz divisors are 2x, 2x + 2, 2x + 4, in general 2x+2k. Example: p0 = 3, G1(3) = 13, G2(3) = 53, G3(3) = 213. The Collatz divisors and Collatz iterates of 3, 13, 53, 213 are 3 (2) 5, 13 (8) 5; 53 (32) 5, 213 (128) 5.
Successive iterates of G(p0) have a cyclic congruence (mod 3): if p0 ≡ 0 (mod 3), then G1(p0) ≡ 1 (mod 3), G2(p0) ≡ 2 (mod 3), G3(p0) ≡ 0 (mod 3) etc. In the above example: 3 ≡ 0 (mod 3), 13 ≡ 1 (mod 3), 53 ≡ 2 (mod 3), 213 ≡ 0 (mod 3).

To calculate all values am,d the program iterates the function G(p) = 4p + 1 using 6 different (basic) start values, the odds: 1, 3, 7, 9, 11 and 17.
The odd 1 is the least odd which is a multiple-of-3 + 1 having the least Collatz divisor d = 2x for x is even (am,d = a1,4) = 1).
The odd 3 is the least odd which is a multiple-of-3 having the least Collatz divisor d = 2x for x is odd (am,d = a0,2 = 3).
The odd 7 is the least odd which is a multiple-of-3 + 1 having the least Collatz divisor d = 2x for x is odd (am,d = a1,2 = 7).
The odd 9 is the least odd which is a multiple-of-3 having the least Collatz divisor d = 2x for x is even (am,d = a0,4 = 9).
The odd 11 is the least odd which is a multiple-of-3 + 2 having the least Collatz divisor d = 2x for x is odd (am,d = a2,2 = 11).
The odd 17 is the least odd which is a multiple-of-3 + 2 having the least Collatz divisor d = 2x for x is even (am,d = a2,4 = 17).

Example 1: start value p = a0,2 = 3, then G1(3) = a1,8 = 13, G2(3) = a2,32 = 53, G3(3) = a0,128 = 213.

Example 2: When G(p) is iterated with start value = 17, Gk(17) is the least odd am,d for m = (17 + k) mod 3, d = 22 + 2k

Example 3: Calculation of am,d for m = 0 and d = 512: d = 512 = 29, 9 = 1 + 2k, k = 4, m = (astart + k) mod 3 = (astart + 4) mod 3 = 0, astart mod 3 = 2. So the start value astart must have x is odd = 1 (d = 2) and astart mod 3 = 2. Start value = a2,2 = 11. Check: a2,2 = 11, G1(11) = 45, G2(11) = 181, G3(11) = 725, G4(11) = 2901. These odds have, successively, a Collatz divisor d = 2, 8, 32, 128, 512 and m = 2, 0, 1, 2, 0. a0,512 = 2901; 2901 (512) 17.

Discussion of the method of calculation
p0, p1 ∈ N+ ∧ p0, p1 is odd, x ∈ N+, p1 = (3p0 + 1) / 2x, G(p0) = 4p0 + 1, p0 ≡ m (mod 3) ⇒ 3G1(p0) + 1 = 3(4p0 + 1) = 4(3p0 + 1) = p1.2x + 2 ⇔ (3G1(p0) + 1) / 2x + 2 = p1
In words: if 2x is the Collatz divisor of p0 then 2x + 2 is the Collatz divisor of G1(p0) and (in reverse) if G1(p0) has the Collatz divisor = 2x + 2 then there is an odd p0 having the Collatz divisor = 2x. The second part of this sentence implies that iteration of a least odd having a Cd (Collatz divisor) = 2x results in a least odd having a Cd = 2x + 2. In a similar way it is proved that in each iteration of G(p0) the Cd of the iterate is multiplied by 4 (the exponent of 2 is increased with 2).
Let p0 = multiple-of-3 + m ⇔ p0 ≡ m (mod 3) ⇒ 4p0 + 1 = 4(multiple-of-3 + m) + 1 = multiple-of-3 + 4m + 1, let m' = 4m + 1 ⇒ if m = 0; m' = 1, if m = 1; m' = 5 mod 3 = 2; if m = 2; m' = 9 mod 3 = 0 ⇔ m' = (m + 1) mod 3. G1(p0) ≡ m' (mod3).
In a similar way it is proved that in each iteration of G(p0) the congruence (mod 3) is increased by 1 (mod 3) and (in reverse) if an iterate pk has the Collatz divisor d = 2y + 2 ∧ pk ≡ m' (mod 3) then there is a pk - 1 having Cd = 2y ∧ pk - 1 ≡ m (mod 3) | m = (m' - 1) mod 3.

The sequences
The program calculates 2 ordered sets (A and B) of positive odd integers. Set A is the sequence of least odds of Collatz classes ordered by the size of these odds and B: the set of the same odds ordered by the size of their Collatz divisors. Set B is a rearrangement of sequence A which makes it easier to see the subsequences of A.
Sequence A = {1,3,5,7,9,11,13,17,21,29,...}.
Sequence A has 3 subsequences: A = A0 ∪ A1 ∪ A2. (after union: order by the size of the terms)
A0 = {3,9,21,45,69,117,213,597,1365,2901,...}; A1 = {1,7,13,37,85,181,277,469,853,2389, ...} and A2 = {5,11,17,29,53,149,341,725,1109,1877,...}.
All terms of A0 are multiples-of-3. All terms of A1 are multiples-of-3 + 1 and all terms of A2 are multiples-of-3 + 2.
Each of the subsequences A0, A1, A2 is the union of 2 other sequences:
A0 = A0O ∪ A0E; A1 = A1O ∪ A1E; A2 = A2O ∪ A2E (after union: order by the size of the terms)
A0O = {3,45,117,213,2901,7509,13653,185685,480597,873813,...}; A0O has terms which are multiples-of-3 having a Collatz divisor d = 2x in which x is an odd exponent. It is the sequence of least odds am,d having m = 0, d = 2x with increasing odd x.
A0E = {9,21,69,597,1365,4437,38229,87381,283989,2446677,...}; A0E has terms which are multiples-of-3 having a Collatz divisor d = 2x in which x is an even exponent. It is the sequence of least odds am,d having m = 0, d = 2x with increasing even x.
A1O = {7,13,181,469,853,11605,30037,54613,742741,1922389,...}; A1O has terms which are multiples-of-3 + 1 having a Collatz divisor d = 2x in which x is an odd exponent. It is the sequence of least odds am,d having m = 1, d = 2x with increasing odd x.
A1E = {1,37,85,277,2389,5461,17749,152917,349525,1135957,...}; A1E has terms which are multiples-of-3 + 1 having a Collatz divisor d = 2x in which x is an even exponent. It is the sequence of least odds am,d having m = 1, d = 2x with increasing even x.
A2O = {11,29,53,725,1877,3413,46421,120149,218453,2970965,...}; A2O has terms which are multiples-of-3 + 2 having a Collatz divisor d = 2x in which x is an odd exponent. It is the sequence of least odds am,d having m = 2, d = 2x with increasing odd x.
A2E = {5,17,149,341,1109,9557,21845,70997,611669,1398101,...} A2E has terms which are multiples-of-3 + 2 having a Collatz divisor d = 2x in which x is an even exponent. It is the sequence of least odds am,d having m = 2, d = 2x with increasing even x.
In the output of sequence A the program gives m of each term and the exponent x of its Collatz divisor, so the above sequences can be read from it. You can also use the output of set B (the rearranged terms of sequence A, ordered now by the size of the Collatz divisors) to find the above sequences.

Definitions
Collatz sequence: pi ∈ N+ ∧ pi is odd, for i > 0: pi = (3pi - 1 + 1) / di - 1. The Collatz divisor di - 1 = 2xi - 1, in which xi - 1 is the number of times that (3p i - 1 + 1) is divided by 2 to reach the odd pi.
Notation: p0 (d0) p1 (d1) p2 (d2) ... pn - 1 (dn - 1) pn | pn = 1 (if the Collatz conjecture is true).
Example: 3 (2) 5 (16) 1

Collatz divisor: The Collatz divisor of an odd p is d = 2x, in which x is the number of times that (3p + 1) can be divided by 2 to reach another odd integer. Examples: 1 (4) (: 1 has the Collatz divisor 4); 3 (2); 5 (16); 7 (2); 9 (4); 11 (2); 13 (8). See the notation of Collatz sequence (above) in which the Collatz divisor separates successive sequence odds.

Collatz iterate: An odd integer pi + 1 calculated from the odd integer pi using the iterative Collatz procedure pi + 1 = (3pi + 1) / di, in which di = 2xi is the Collatz divisor of pi.

Collatz class: The Collatz class C(m,d) is the set of odd positive integers p having an equal congruence (mod 3) = m and an equal Collatz divisor d.
Example: C(0,128) is the set of all positive odd multiples of 3 having a Collatz divisor of 128 = {213, 981, 1749, 2517, 3285, ...}: 213 (128) 5; 981 (128) 23; 1749 (128) 41; etc.
In general C(m,d) = {p | p = am,d + h * 6 * d} in which am,d is the least odd of the set and h ∈ {0, 1, 2, ...}. In the sample above C(0,128) = {p | p = 213 + h *6 *128}.
The iterates q of the elements p of C(m,d) = {q | q = q0 + h * 2 * 32}, in which q0 is the iterate of am,d.

Other relevant programs on this site
If you click a relevant program name down here, you will arrive at the CCCC Index Page, where the program is described. To come back here click 'previous' or select the name of the present program (Sequence of least odds of Collatz classes and Comment) at the CCCC Index Page or at the CCCC Contents Page.