Math::NumSeq::DigitProductSteps -- multiplicative persistence and digital root


 use Math::NumSeq::DigitProductSteps;
 my $seq = Math::NumSeq::DigitProductSteps->new (values_type => 'count');
 my ($i, $value) = $seq->next;


This is an iteration taking the product of the digits of a number until reaching a single digit value. The sequence values are the count of steps, also called the multiplicative persistence.

    starting i=0

For example i=39 goes 3*9=27 -> 2*7=14 -> 1*4=4 to reach a single digit, so value=3 iterations.

The values_type => 'root' gives the final digit reached by the steps, which is called the multiplicative digital root.

    values_type => 'root'

i=0 through i=9 are already single digits so their count is 0 and root is the value itself. Then i=10 to i=19 all take just a single iteration to reach a single digit. i=25 is the first to require 2 iterations.

Any i with a 0 digit takes just one iteration to get to root 0. Any i like 119111 which is all 1s except for at most a single non-1 takes just one iteration. This includes the repunits 111..11.


An optional radix parameter selects a base other than decimal.

Binary radix=>2 is not very interesting since the digit product is always either 0 or 1. i>=2 always takes just 1 iteration and has root 0 except for i=2^k-1 all 1s with root 1.


See "FUNCTIONS" in Math::NumSeq for behaviour common to all sequence classes.

$seq = Math::NumSeq::DigitProductSteps->new ()
$seq = Math::NumSeq::DigitProductSteps->new (values_type => $str, radix => $integer)

Create and return a new sequence object.

Random Access

$value = $seq->ith($i)

Return the sequence value, either count or final root value as selected.

$bool = $seq->pred($value)

Return true if $value occurs in the sequence. For the count of steps this means any integer $value >= 0, or for a root any digit 0 <= $value < radix.


Math::NumSeq, Math::NumSeq::DigitProduct



Copyright 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 <>.