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 modified Collatz-Terras map (A060322), called in the Vaillant and Delarue link f.

The Collatz conjecture: iterations of the map f = a: N_0 -> N_0 with n -> a(n) lead always to 0.

The minimal number k with a^{[k]}(n) = 0 is given by A324037(n).

The tree CfTree, related to this map, giving the branches which lead to 0 for each vertex label of level n >= 0 is given in A324246.

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.

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

The branches of this incomplete binary tree, called CfsTree, give the iterations of the vertex labels of level n of the modified reduced Collatz map Cfs defined by Cfs(2*k+1) = A324036(k), for k >= 0, until at level n = 0 the label 1 is reached for the first time.

The out-degree of a vertex label T(n, k), for n >= 1, is 1 if T(n, k) == 3 (mod 6) and 2 for all other vertices. For level n = 0 with vertex label 1 this rule does not hold, it has out-degree 1, not 2.

The number of vertex labels on level n which are 3 (mod 6) is given by A324040(n).

The corresponding tree with nonnegative vertex labels t(n, k) = (T(n,k) - 1)/2 is given in A324246.

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

For this tree see Figure 1 in the Vaillant-Delarue link. It is also shown in the W. Lang link.

The Collatz conjecture is that a(n) is finite. If 1 should never be reached then a(n) = -1.

Compare this sequence with the analogous one A075680(n+1) for the reduced Collatz map of A075677.

a(n) gives also the minimal number of iterations of the Vaillant-Delarue map f, defined in A324245, acting on n to reach 0 (assuming the Collatz conjecture).

For the link to the Vaillant-Delarue paper (where fs is called f_s) see A324036.

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).