1Math::NumSeq::CollatzStUespesr(3C)ontributed Perl DocumeMnattaht:i:oNnumSeq::CollatzSteps(3)
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6 Math::NumSeq::CollatzSteps -- steps in the "3n+1" problem
7
9 use Math::NumSeq::CollatzSteps;
10 my $seq = Math::NumSeq::CollatzSteps->new;
11 my ($i, $value) = $seq->next;
12
14 The number of steps it takes to reach 1 by the Collatz "3n+1" problem,
15 0, 1, 7, 2, 5, 8, 16, 3, 19, 6, 14, 9, 9, 17, 17, 4, 12, 20, ...
17 starting i=1
18 The Collatz problem iterates
20 n -> / 3n+1 if n odd
22 \ n/2 if n even
23 For example i=6 takes value=8 many steps to reach 1,
25 6 -> 3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1
27 It's conjectured that any starting n will always eventually reduce to 1
29 and so the number of steps is finite. There's no limit in the code on
30 how many steps counted. "Math::BigInt" is used if 3n+1 steps go past
31 the usual scalar integer limit.
32 Up Steps
34 Option "step_type => "up"" counts only the 3n+1 up steps.
35 step_type => "up"
37 0, 0, 2, 0, 1, 2, 5, 0, 6, 1, 4, 2, 2, 5, 5, 0, 3, 6, 6, 1, 1, ...
38 This can also be thought of as steps iterating
40 n -> (3n+1)/2^k for maximum k
42 Down Steps
44 Option "step_type => "down"" counts only the n/2 down steps.
45 step_type => "down"
47 0, 1, 5, 2, 4, 6, 11, 3, 13, 5, 10, 7, 7, 12, 12, 4, 9, 14, ...
48 The total up+down gives the default "step_type=both".
50 Odd Numbers
52 Option "on_values => "odd"" counts steps on the odd numbers 2*i+1.
53 on_values => "odd"
55 0, 7, 5, 16, 19, 14, 9, 17, 12, 20, 7, 15, 23, 111, 18, 106, ...
56 starting i=0 for odd number 1
57 Even Numbers
59 Option "on_values => "even"" counts steps on the even number 2*i,
60 on_values => "even"
62 1, 2, 8, 3, 6, 9, 17, 4, 20, 7, 15, 10, 10, 18, 18, 5, 13, 21, ...
63 starting i=0 for even number 2
64 Since 2*i is even the first step is down n/2 to give i and thereafter
66 the same as the plain count. This means the steps for "even" is simply
67 1 more than for plain "all".
68
70 See "FUNCTIONS" in Math::NumSeq for behaviour common to all sequence
71 classes.
72 "$seq = Math::NumSeq::CollatzSteps->new ()"
74 "$seq = Math::NumSeq::CollatzSteps->new (step_type => $str, on_values
75 => $str)"
76 Create and return a new sequence object. The optional "step_type"
77 parameter (a string) can be
78 "up" upward steps 3n+1
80 "down" downward steps n/2
81 "both" both up and down (the default)
82 The optional "on_values" parameter (a string) can be
84 "all" all integers i
86 "odd" odd integers 2*i+1
87 "even" even integers 2*i
88 Random Access
90 "$value = $seq->ith($i)"
91 Return the number of steps to take $i down to 1.
92 "$bool = $seq->pred($value)"
94 Return true if $value occurs as a step count. This is simply
95 "$value >= 0".
96
98 Math::NumSeq, Math::NumSeq::JugglerSteps, Math::NumSeq::ReverseAddSteps
99 Math::NumSeq::HappySteps
100
102 <http://user42.tuxfamily.org/math-numseq/index.html>
103
105 Copyright 2011, 2012, 2013, 2014, 2016, 2019 Kevin Ryde
106 Math-NumSeq is free software; you can redistribute it and/or modify it
108 under the terms of the GNU General Public License as published by the
109 Free Software Foundation; either version 3, or (at your option) any
110 later version.
111 Math-NumSeq is distributed in the hope that it will be useful, but
113 WITHOUT ANY WARRANTY; without even the implied warranty of
114 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
115 General Public License for more details.
116 You should have received a copy of the GNU General Public License along
118 with Math-NumSeq. If not, see <http://www.gnu.org/licenses/>.
119perl v5.32.0 2020-07-28 Math::NumSeq::CollatzSteps(3)