cp's OEIS Frontend

The length of row n is A324039, for n >= 0.

This is the incomplete binary tree corresponding to the modified Collatz map f (from the Vaillant and Delarue link) given in A324245.

The branches of this tree, called CfTree, give the iterations under the Vaillant and Delarue map f of the vertex labels of level n until label 0 on level n = 0 is reached.

The out-degree of a vertex label T(n, k), for n >= 1, is 1 if T(n, k) == 1 (mod 3) and 2 for all other labels. For level n = 0 with vertex label 0 this rule does not hold, it has out-degree 1, not 2.

The number of vertex labels on level n which are 1 (mod 3) is given in A324040.

The corresponding tree CfsTree with only odd vertex labels t(n, k) = 2*T(n,k) + 1 is given in A324038.

The Collatz conjecture is that all nonnegative integers appear in this CfTree. Because the sets of labels on the levels are pairwise disjoint, these numbers will then appear just once.

For this tree see Figure 2 in the Vaillant-Delarue link. It is also shown in the W. Lang link given in A324038.

This is a modification of the reduced Collatz map given in A075677.

The Collatz conjecture is that iteration of the map fs leads to 1 for all positive odd integers.

In the Vaillant-Delarue (V-D) reference the present map fs: Odd -> Odd, 2*n+1 -> a(n) = fs(2*n+1), for n >= 0, is called f_{s}. The differences from b(n) = A075677(n+1) = fCr(2*n+1) (called f_{cr} in V-D) occur for the positions n = 2 + 4*k, for k >= 1: b(2 + 4*k) = b(k) = A075677(k+1) but a(2 + 4*k) = 1 + 2*k, which differs.

The advantage of the map fs (or a) over fCr (or b) is an explicit formula over a recurrence.

Additional steps are introduced in the iteration of fs versus fCr. This leads to an incomplete binary tree, called CfsTree, given in A324038. No such tree is available for fCr.

Such additional steps in fs can only occur after odd numbers congruent to 5 modulo 8: fs(5 + 8*k) = a(2 + 4*k) = 1 + 2*k and fs(1 + 2*k) = a(k). On the other hand, fCr(5 + 8*k) = b(2 + 4*k) = b(k).

The appearance of exactly N consecutive steps in fs versus fCr, for N >= 2, can be shown recursively to start with the odd numbers O(N;k) = 1 + 4*O(N-1;k), for N >= 3, with input O(2;k) = 53 + (4^3)*k. These are the numbers O(N;k) = A072197(N) + A000302(N+1)*k, for N >= 2. Therefore only one additional step follows directly after an odd number 5 (mod 8) if it is not of the O(N;k) type for N >= 2.

The minimal number of iterations of function fs acting on 2*n + 1 (or a acting on n), for n >= 0, to reach 1 is given in A324037 (if for very large n the number 1 should not be reached A324037(n) is set to -1).

a(n) is also the number of vertex labels congruent to 3 modulo 6 of row n of the irregular triangle A324038.

This entry is interesting because it determines the number of vertices with out-degree 1 of level n, for n >= 1, of the modified reduced Collatz trees A324038 and A324246. All other vertices have out-degree 2. Hence this sequence determines recursively the number A324039(n) of vertices of label n of these two trees.