# NAME

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

# SYNOPSIS

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

# DESCRIPTION

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.

``````    0,0,..0,0,1,1,..1,1,2,2,2,2,2,1,1,1,1,2,2,2,2,2,3,1,1,1,2,...
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'
0,1,2,...,9,0,1,...,9,0,2,4,6,8,0,2,4,6,8,0,3,6,9,2,5,8,...``````

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.

# FUNCTIONS

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`.