OFFSET
1,13
COMMENTS
a(n) is the number of tuples (t(1), ..., t(n)) of positive integers t(i) such that t(i+1) = t(i) / 2 or t(i+1) = 3*t(i) + 1 for all i < n, t(1) = t(n) / 2 or t(1) = 3*t(n) + 1, t(i) <> t(j) for i <> j, and t(1) < t(i) for all i > 1.
These tuples correspond to cycles of a "pseudo-Collatz" iteration that allows tripling steps for even numbers as well. Only "prime" cycles are allowed, i.e., cycles that cannot be composed from shorter cycles, which is equivalent to the uniqueness of each entry.
Concerning the correctness of the PARI program: The program checks only that one element of a discovered cycle is integer. This is sufficient, for suppose t(i) is integer and its successor t(j) is not. Then t(j) = k + 1/2 with an integer number k. When a rational number s is not integer and has a finite binary expansion, then the same is true for s/2 and for 3s+1. Repeated application of this leads to a contradiction.
The results up to n = 45 and an extended search up to n = 60 suggest that, apart from the trivial cycle, no pseudo-Collatz cycle exists for which each 3x+1 step is followed by an x/2 step. A proof of this would imply the "no nontrivial cycle" part of the Collatz conjecture.
LINKS
EXAMPLE
There is one such tuple for n = 3, namely (1,4,2), hence a(3) = 1.
There is one such tuple for n = 8, namely (4,13,40,20,10,5,16,8), hence a(8) = 1.
There are six such tuples for n = 13, namely (13,40,20,61,184,92,277,832,416,208,104,52,26), (14,43,130,65,196,98,49,148,74,37,112,56,28), (16,49,148,74,37,112,56,28,85,256,128,64,32), (19,58,29,88,44,22,67,202,101,304,152,76,38), (19,58,29,88,44,133,400,200,100,50,25,76,38), (20,61,184,92,46,23,70,35,106,53,160,80,40), hence a(13) = 6.
PROG
(PARI) \\ see link. a354883(n) prints and counts all admitted tuples of length n and returns the number of tuples.
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Markus Sigg, Jun 10 2022
STATUS
approved