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

16, 4, 5, 1, 10, 2, 3, 40, 12, 13, 64, 20, 21, 88, 28, 29, 9, 58, 112, 36, 37, 136, 44, 45, 160, 52, 53, 17, 106, 34, 35, 11, 70, 22, 23, 7, 46, 14, 15, 184, 60, 61, 208, 68, 69, 232, 76, 77, 25, 154, 50, 51, 256, 84, 85, 280, 92, 93

Offset: 1

Author

Georg Fischer, Apr 14 2019

Keywords

• nonn
• easy
• tabf

Comments

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.

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".

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:

k j=4*k+2 j=4*k+3 j=4*k+4 j=4*k+5

--------------------------------------------------

0 mm dmm mmd dmmd

1 mms dmms mmsd dmmsd

2 mms^2 dmms^2 mms^2d dmms^2d

...

k mms^k dmms^k mm(s^k)d dmm(s^k)d

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.

Conjecture: The sequence is a permutation of the positive numbers.

Examples

			Table S(i, j) begins:
  i\j    1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
  ----------------------------------------------------------------
  1:    16   4   5   1  10   2   3
  2:    40  12  13
  3:    64  20  21
  4:    88  28  29   9  58
  5:   112  36  37
  6:   136  44  45
  7:   160  52  53  17 106  34  35  11  70  22  23   7  46  14  15
  8:   184  60  61
		

Links

• Georg Fischer, Perl program for the generation of related sequences.
• Manfred Trümper, The Collatz Problem in the Light of an Infinite Free Semigroup, Chinese Journal of Mathematics, Vol. 2014, Article ID 756917, 21 pages.

Crossrefs

Cf. A160016 (level 3), A307048 (level 2), A322469 (level 1).

Programs

Perl

cf. link.