user
Definition

The **radical** of a positive integer is defined as the product of its distinct prime factors.

Although the integer 1 has no prime factors, its radical is regarded as 1 by convention.

Example

The radical of 504 = 2³ x 3² x 7 is: 2 x 3 x 7 = 42.

Task

1. Find and show on this page the radicals of the first **50** positive integers.

2. Find and show the radicals of the integers: **99999, 499999** and **999999**.

3. By considering their radicals, show the distribution of the first **one million** positive integers by numbers of distinct prime factors (hint: the maximum number of distinct factors is 7).

Bonus

By (preferably) using an independent method, calculate the number of primes and the number of powers of primes less than or equal to **one million** and hence check that your answer in 3. above for numbers with one distinct prime is correct.

Related tasks

* Square-free integers
* Almost prime

References

* Wikipedia article Radical of an integer
* OEIS sequence A007947: Largest square free number dividing n
assistant
```
use Prime::Factor;
use List::Divvy;
use Lingua::EN::Numbers;

sub radical ($_) { [×] unique .&prime-factors }

say "First fifty radicals:\n" ~
  (1..50).map({.&radical}).batch(10)».fmt("%2d").join: "\n";
say '';

printf "Radical for %7s => %7s\n", .&comma, comma .&radical
  for 99999, 499999, 999999;

my %rad = 1 => 1;
my $limit = 1e6.Int;

%rad.push: $_ for (2..$limit).race(:1000batch).map: {(unique .&prime-factors).elems => $_};

say "\nRadical factor count breakdown, 1 through {comma $limit}:";
say .key ~ " => {comma +.value}" for sort %rad;

my @primes = (2..$limit).grep: &is-prime;

my int $powers;
@primes.&upto($limit.sqrt.floor).map: -> $p {
   for (2..*) { ($p ** $_) < $limit ?? ++$powers !! last }
}

say qq:to/RADICAL/;

    Up to {comma $limit}:
    Primes: {comma +@primes}
    Powers:    $powers
    Plus 1:      1
    Total:  {comma 1 + $powers + @primes}
    RADICAL

```

Output:

```
First fifty radicals:
 1  2  3  2  5  6  7  2  3 10
11  6 13 14 15  2 17  6 19 10
21 22 23  6  5 26  3 14 29 30
31  2 33 34 35  6 37 38 39 10
41 42 43 22 15 46 47  6  7 10

Radical for  99,999 =>  33,333
Radical for 499,999 =>   3,937
Radical for 999,999 => 111,111

Radical factor count breakdown, 1 through 1,000,000:
1 => 78,735
2 => 288,726
3 => 379,720
4 => 208,034
5 => 42,492
6 => 2,285
7 => 8

Up to 1,000,000:
Primes: 78,498
Powers:    236
Plus 1:      1
Total:  78,735

```