cp's OEIS Frontend

A307407 Irregular table read by rows: rows list terms that map to the nodes in the graph of the "3x+1" (or Collatz) problem.

%I A307407 #14 Jun 04 2021 10:51:52
%S A307407 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 A307407 160,52,53,17,106,34,35,11,70,22,23,7,46,14,15,184,60,61,208,68,69,
%U A307407 232,76,77,25,154,50,51,256,84,85,280,92,93
%N A307407 Irregular table read by rows: rows list terms that map to the nodes in the graph of the "3x+1" (or Collatz) problem.
%C A307407 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 A307407 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 A307407 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 A307407   k   j=4*k+2    j=4*k+3    j=4*k+4    j=4*k+5
%C A307407   --------------------------------------------------
%C A307407   0   mm         dmm        mmd        dmmd
%C A307407   1   mms        dmms       mmsd       dmmsd
%C A307407   2   mms^2      dmms^2     mms^2d     dmms^2d
%C A307407   ...
%C A307407   k   mms^k      dmms^k     mm(s^k)d   dmm(s^k)d
%C A307407 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 A307407 Conjecture: The sequence is a permutation of the positive numbers.
%H A307407 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 A307407 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 A307407 Table S(i, j) begins:
%e A307407   i\j    1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
%e A307407   ----------------------------------------------------------------
%e A307407   1:    16   4   5   1  10   2   3
%e A307407   2:    40  12  13
%e A307407   3:    64  20  21
%e A307407   4:    88  28  29   9  58
%e A307407   5:   112  36  37
%e A307407   6:   136  44  45
%e A307407   7:   160  52  53  17 106  34  35  11  70  22  23   7  46  14  15
%e A307407   8:   184  60  61
%o A307407 (Perl) cf. link.
%Y A307407 Cf. A160016 (level 3), A307048 (level 2), A322469 (level 1).
%K A307407 nonn,easy,tabf
%O A307407 1,1
%A A307407 _Georg Fischer_, Apr 14 2019