Math::NumSeq::JugglerSteps -- steps in the juggler sqrt sequence
use Math::NumSeq::JugglerSteps; my $seq = Math::NumSeq::JugglerSteps->new; my ($i, $value) = $seq->next;
This is the number of steps it takes to reach 1 by the Juggler sqrt sequence,
n -> / floor(n^(1/2)) if n even \ floor(n^(3/2)) if n odd
So sqrt if n even, or sqrt(n^3) if n odd, each rounded downwards. For example i=17 goes 17 -> sqrt(17^3)=70 -> sqrt(70)=8 -> sqrt(8)=2 -> sqrt(2)=1, for a count of 4 steps.
0, 1, 6, 2, 5, 2, 4, 2, 7, 7, 4, 7, 4, 7, 6, 3, 4, 3, 9, 3, ... starting i=1
The intermediate values in the calculation can become quite large and Math::BigInt
is used if necessary. There's some secret experimental caching in a temporary file, for a small speedup.
See "FUNCTIONS" in Math::NumSeq for behaviour common to all sequence classes.
$seq = Math::NumSeq::JugglerSteps->new ()
$seq = Math::NumSeq::JugglerSteps->new (step_type => 'both', juggler_type => '1/2-3/2')
Create and return a new sequence object.
The optional step_type
parameter (a string) selects between
"up" upward steps sqrt(n^3) "down" downward steps sqrt(n) "both" both up and down, which is the default
The optional juggler_type
parameter (a string) selects among variations on the powering
n even n odd "1/2-3/2" sqrt(n) and sqrt(n^3) "2/3-3/2" cbrt(n^2) and sqrt(n^3) "3/4-4/3" n^(3/4) and n^(4/3)
$value = $seq->ith($i)
Return the number of steps to take $i
down to 1.
$bool = $seq->pred($value)
Return true if $value
occurs as a step count. This is simply $value >= 0
.
Math::NumSeq, Math::NumSeq::CollatzSteps, Math::NumSeq::ReverseAddSteps
http://user42.tuxfamily.org/math-numseq/index.html
Copyright 2010, 2011, 2012, 2013, 2014 Kevin Ryde
Math-NumSeq is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version.
Math-NumSeq is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.
You should have received a copy of the GNU General Public License along with Math-NumSeq. If not, see <http://www.gnu.org/licenses/>.