Sequence A006877: 3X+1 Record-Setters (Number of Steps)
(The 3X+1 sequences are also discussed here.)
This sequence, Sloane's A006877, gives the numbers that "set a new record" for the number of steps of the "3x+1" iteration that are required to reach 1.
A006877: 2, 3, 6, 7, 9, 18, 25, 27, 54, 73, 97, 129, 171, 231, 313, 327, 649, 703, 871, 1161, 2223, 2463, 2919, 3711, 6171, 10971, 13255, 17647, 23529, 26623, 34239, 35655, 52527, 77031, 106239, 142587, 156159, 216367, 230631, 410011, 511935, 626331, 837799, 1117065, 1501353, 1723519, 2298025, 3064033, 3542887, 3732423, 5649499, 6649279, 8400511, 11200681, 14934241, 15733191, 31466382, 36791535, 63728127, 127456254, 169941673, 226588897, 268549803, 537099606, 670617279, 1341234558, ...
The "3x+1 problem", more formally known as the Collatz conjecture, involves the question of whether the following process will always reach 1 (if the conjecture is true) or might grow forever for certain starting values (if the conjecture is false):
- Choose a starting value and call it X.
- If X is even, divide by 2.
- If X is odd, multiply by 3 and add 1.
- Continue computing X/2 or 3X+1 until X reaches 1.
Some starting values, such as 27, take quite a few steps to reach 1.
The problem is described in the book Goedel,_Escher,_Bach, where it is called the "Wondrous number" question. In 1981 I wrote an AppleSoft BASIC program to generate this sequence:
Generating A6877, A6884, A6878 and A6885 on an Apple ][
I submitted the sequence to N.J.A. Sloane on the 7th January 1993, along with A006844, A006878, and A006885. I described all of them in terms of Hofstadter's description using the name "wondrous", as I did not yet know of the history of the problem.
In the time it takes the Apple ][ to generate the terms up through 327, a modern computer can get as far as 670617279.
This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2024 Mar 28.
s.27