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 = 2^x 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 = a_m,d + h * 6 * d} in which a_m,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 = 2^x. When the least odd a_m,d of the set is known, all p of the set can be calculated. The iterative function the program uses to calculate a_m,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 G^k(p₀) have equal Collatz iterates p₁. The corresponding Collatz divisors are 2^x, 2^{x + 2}, 2^{x + 4}, in general 2^x+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 a_m,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 = 2^x for x is even (a_m,d = a_1,4) = 1).
The odd 3 is the least odd which is a multiple-of-3 having the least Collatz divisor d = 2^x for x is odd (a_m,d = a_0,2 = 3).
The odd 7 is the least odd which is a multiple-of-3 + 1 having the least Collatz divisor d = 2^x for x is odd (a_m,d = a_1,2 = 7).
The odd 9 is the least odd which is a multiple-of-3 having the least Collatz divisor d = 2^x for x is even (a_m,d = a_0,4 = 9).
The odd 11 is the least odd which is a multiple-of-3 + 2 having the least Collatz divisor d = 2^x for x is odd (a_m,d = a_2,2 = 11).
The odd 17 is the least odd which is a multiple-of-3 + 2 having the least Collatz divisor d = 2^x for x is even (a_m,d = a_2,4 = 17).

Example 1: start value p = a_0,2 = 3, then G¹(3) = a_1,8 = 13, G²(3) = a_2,32 = 53, G³(3) = a_0,128 = 213.

Example 2: When G(p) is iterated with start value = 17, G^k(17) is the least odd a_m,d for m = (17 + k) mod 3, d = 2^{2 + 2k}

Example 3: Calculation of a_m,d for m = 0 and d = 512: d = 512 = 2⁹, 9 = 1 + 2k, k = 4, m = (a_start + k) mod 3 = (a_start + 4) mod 3 = 0, a_start mod 3 = 2. So the start value a_start must have x is odd = 1 (d = 2) and a_start mod 3 = 2. Start value = a_2,2 = 11. Check: a_2,2 = 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. a_0,512 = 2901; 2901 (512) 17.

Discussion of the method of calculation
p₀, p₁ ∈ N⁺ ∧ p₀, p₁ is odd, x ∈ N⁺, p₁ = (3p₀ + 1) / 2^x, G(p₀) = 4p₀ + 1, p₀ ≡ m (mod 3) ⇒ 3G¹(p₀) + 1 = 3(4p₀ + 1) = 4(3p₀ + 1) = p₁.2^{x + 2} ⇔ (3G¹(p₀) + 1) / 2^{x + 2} = p₁
In words: if 2^x is the Collatz divisor of p₀ then 2^{x + 2} is the Collatz divisor of G¹(p₀) and (in reverse) if G¹(p₀) has the Collatz divisor = 2^{x + 2} then there is an odd p₀ having the Collatz divisor = 2^x. The second part of this sentence implies that iteration of a least odd having a Cd (Collatz divisor) = 2^x results in a least odd having a Cd = 2^{x + 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 p_k has the Collatz divisor d = 2^{y + 2} ∧ p_k ≡ m' (mod 3) then there is a p_{k - 1} having Cd = 2^y ∧ p_{k - 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 = A_0O ∪ A_0E; A₁ = A_1O ∪ A_1E; A₂ = A_2O ∪ A_2E (after union: order by the size of the terms)
A_0O = {3,45,117,213,2901,7509,13653,185685,480597,873813,...}; A_0O has terms which are multiples-of-3 having a Collatz divisor d = 2^x in which x is an odd exponent. It is the sequence of least odds a_m,d having m = 0, d = 2^x with increasing odd x.
A_0E = {9,21,69,597,1365,4437,38229,87381,283989,2446677,...}; A_0E has terms which are multiples-of-3 having a Collatz divisor d = 2^x in which x is an even exponent. It is the sequence of least odds a_m,d having m = 0, d = 2^x with increasing even x.
A_1O = {7,13,181,469,853,11605,30037,54613,742741,1922389,...}; A_1O has terms which are multiples-of-3 + 1 having a Collatz divisor d = 2^x in which x is an odd exponent. It is the sequence of least odds a_m,d having m = 1, d = 2^x with increasing odd x.
A_1E = {1,37,85,277,2389,5461,17749,152917,349525,1135957,...}; A_1E has terms which are multiples-of-3 + 1 having a Collatz divisor d = 2^x in which x is an even exponent. It is the sequence of least odds a_m,d having m = 1, d = 2^x with increasing even x.
A_2O = {11,29,53,725,1877,3413,46421,120149,218453,2970965,...}; A_2O has terms which are multiples-of-3 + 2 having a Collatz divisor d = 2^x in which x is an odd exponent. It is the sequence of least odds a_m,d having m = 2, d = 2^x with increasing odd x.
A_2E = {5,17,149,341,1109,9557,21845,70997,611669,1398101,...} A_2E has terms which are multiples-of-3 + 2 having a Collatz divisor d = 2^x in which x is an even exponent. It is the sequence of least odds a_m,d having m = 2, d = 2^x 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: p_i ∈ N⁺ ∧ p_i is odd, for i > 0: p_i = (3p_{i - 1} + 1) / d_{i - 1}. The Collatz divisor d_{i - 1} = 2^{x_{i - 1}}, in which x_{i - 1} is the number of times that (3p _{i - 1} + 1) is divided by 2 to reach the odd p_i.
Notation: p₀ (d₀) p₁ (d₁) p₂ (d₂) ... p_{n - 1} (d_{n - 1}) p_n | p_n = 1 (if the Collatz conjecture is true).
Example: 3 (2) 5 (16) 1

Collatz divisor: The Collatz divisor of an odd p is d = 2^x, 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 p_{i + 1} calculated from the odd integer p_i using the iterative Collatz procedure p_{i + 1} = (3p_i + 1) / d_i, in which d_i = 2^x_i is the Collatz divisor of p_i.

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 = a_m,d + h * 6 * d} in which a_m,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 a_m,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.

Collatz classes
Some relations of positive odds, Collatz divisors and first Collatz iterates and Comment
A simple tree structure of Collatz sequences
Composite Collatz classes (continued) and Comment