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)

Random Access

$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



Copyright 2010, 2011, 2012, 2013, 2014, 2016, 2019, 2020 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 <>.