|
%I #14 Jun 04 2021 10:51:52
%S 16,4,5,1,10,2,3,40,12,13,64,20,21,88,28,29,9,58,112,36,37,136,44,45,
%T 160,52,53,17,106,34,35,11,70,22,23,7,46,14,15,184,60,61,208,68,69,
%U 232,76,77,25,154,50,51,256,84,85,280,92,93
%N Irregular table read by rows: rows list terms that map to the nodes in the graph of the "3x+1" (or Collatz) problem.
%C The construction is similar to that in A322469. The sequence is the flattened form of an irregular table S(i, j) (see the example below) which has rows i >= 1 consisting of subsequences of varying length.
%C Like Truemper (cf. link), we denote the mapping x -> 2*x by "m" ("multiply"), the mapping x -> (x - 1)/3 by "d" ("divide"), and the combined mapping "dm" x -> (x - 1)/3 * 2 by "s" ("squeeze"). The d mapping is defined only if it is possible, that is, if x - 1 is divisible by 3. We write m, d and s as infix operation words, for example "4 mms 10", and we use exponents for repeated operations, for example "mms^2 = mmss".
%C Row i in table S is constructed by the following algorithm: Start with 6 * i - 2 in column j = 1. The following columns j are defined in groups of four by the operations:
%C k j=4*k+2 j=4*k+3 j=4*k+4 j=4*k+5
%C --------------------------------------------------
%C 0 mm dmm mmd dmmd
%C 1 mms dmms mmsd dmmsd
%C 2 mms^2 dmms^2 mms^2d dmms^2d
%C ...
%C k mms^k dmms^k mm(s^k)d dmm(s^k)d
%C The construction for the row terminates at the first column where a d operation is no longer possible. This point is always reached. This can be proved by the observation that, for any row i in S, there is a unique mapping x -> (x + 2)/6 of the terms in column 1, 2, 5, 9, 13, ... 4*m+1 to the terms in row i of table T in A322469. The row construction process in A322469 stops, therefore it stops also in the sequence defined here.
%C Conjecture: The sequence is a permutation of the positive numbers.
%H Georg Fischer, <a href="https://github.com/gfis/fasces/blob/master/oeis/collatz/segment.pl">Perl program for the generation of related sequences</a>.
%H Manfred Trümper, <a href="http://dx.doi.org/10.1155/2014/756917">The Collatz Problem in the Light of an Infinite Free Semigroup</a>, Chinese Journal of Mathematics, Vol. 2014, Article ID 756917, 21 pages.
%e Table S(i, j) begins:
%e i\j 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
%e ----------------------------------------------------------------
%e 1: 16 4 5 1 10 2 3
%e 2: 40 12 13
%e 3: 64 20 21
%e 4: 88 28 29 9 58
%e 5: 112 36 37
%e 6: 136 44 45
%e 7: 160 52 53 17 106 34 35 11 70 22 23 7 46 14 15
%e 8: 184 60 61
%o (Perl) cf. link.
%Y Cf. A160016 (level 3), A307048 (level 2), A322469 (level 1).
%K nonn,easy,tabf
%O 1,1
%A _Georg Fischer_, Apr 14 2019
|
