# NAME Smooth::Numbers

# SYNOPSIS

Given a list of prime factors, return an iterator which will produce numbers that are "smooth" to those factors.

```
use Smooth::Numbers;
my $Hammings = smooth-numbers(2,3,5);
put $Hammings[^20]; # 1 2 3 4 5 6 8 9 10 12 15 16 18 20 24 25 27 30 32 36
```

# DESCRIPTION

Smooth numbers are defined as "numbers that are products of small prime factors."

Hamming numbers, which are the numbers that are products of 2, 3, 5, is the classic example.

The first ten Hamming numbers ( 2,3,5 smooth numbers ) are:

```
1 ( 2⁰ * 3⁰ * 5⁰ )
2 ( 2¹ * 3⁰ * 5⁰ )
3 ( 2⁰ * 3¹ * 5⁰ )
4 ( 2² * 3⁰ * 5⁰ )
5 ( 2⁰ * 3⁰ * 5¹ )
6 ( 2¹ * 3¹ * 5⁰ )
8 ( 2³ * 3⁰ * 5⁰ )
9 ( 2⁰ * 3² * 5⁰ )
10 ( 2¹ * 3⁰ * 5¹ )
12 ( 2² * 3¹ * 5⁰ )
```

Smooth numbers are easy to conceptualize but can be challenging to generate in order. This module provides a simple routine to overcome that.

Exports a single subroutine:

```
sub smooth-numbers(*@factors)
```

Takes a list or array of factors and returns an iterator that generates an ordered sequence of numbers that are smooth to those factors.

The factors need not be in any order, and need not be prime. Smooth numbers traditionally are thought of as products of only prime factors, but if you want, for instance, the products of powers of 4 and 9, this module will oblige.

```
put smooth-numbers(4,9)[^15];
# 1 4 9 16 36 64 81 144 256 324 576 729 1024 1296 2304
```

For that matter, there is no constraint that the "factors" need even be integers.

```
put smooth-numbers(1.5,2.5)[^10];
# 1 1.5 2.25 2.5 3.375 3.75 5.0625 5.625 6.25 7.59375
```

There *are* a few constraints:

there must be at least one passed factor.

the passed parameters must all be numeric. (Can't exponentiate strings).

the factors must all be greater than or equal to 1 (or the sequence would be non-ascending).

Note that if *any* of the factors is equal to 1, the sequence will return an infinite series of 1s. Should be obvious why if you think about it.

# AUTHOR

2019 Steve Schulze aka thundergnat

This package is free software and is provided "as is" without express or implied warranty. You can redistribute it and/or modify it under the same terms as Perl itself.

# LICENSE

Licensed under The Artistic 2.0; see LICENSE.