# Square-free integers

In Sidef, the functions is_square_free(n) and square_free_count(min, max) are built-in. However, we can very easily reimplement them in Sidef code, as fast integer factorization methods are also available in the language.

``````func is_square_free(n) {

n.abs!       if (n <  0)
return false if (n == 0)

n.factor_exp + [[1,1]] -> all { . == 1 }
}

func square_free_count(n) {
1 .. n.isqrt -> sum {|k|
moebius(k) * idiv(n, k*k)
}
}

func display_results(a, c, f = { _ }) {
a.each_slice(c, {|*s|
say s.map(f).join(' ')
})
}

var a = range(   1,      145).grep {|n| is_square_free(n) }
var b = range(1e12, 1e12+145).grep {|n| is_square_free(n) }

say "There are #{a.len} square─free numbers between 1 and 145:"
display_results(a, 17, {|n| "%3s" % n })

say "\nThere are #{b.len} square─free numbers between 10^12 and 10^12 + 145:"
display_results(b, 5)
say ''

for (2 .. 6) { |n|
var c = square_free_count(10**n)
say "The number of square─free numbers between 1 and 10^#{n} (inclusive) is: #{c}"
}
``````

#### Output:

``````There are 90 square─free numbers between 1 and 145:
1   2   3   5   6   7  10  11  13  14  15  17  19  21  22  23  26
29  30  31  33  34  35  37  38  39  41  42  43  46  47  51  53  55
57  58  59  61  62  65  66  67  69  70  71  73  74  77  78  79  82
83  85  86  87  89  91  93  94  95  97 101 102 103 105 106 107 109
110 111 113 114 115 118 119 122 123 127 129 130 131 133 134 137 138
139 141 142 143 145

There are 89 square─free numbers between 10^12 and 10^12 + 145:
1000000000001 1000000000002 1000000000003 1000000000005 1000000000006
1000000000007 1000000000009 1000000000011 1000000000013 1000000000014
1000000000015 1000000000018 1000000000019 1000000000021 1000000000022
1000000000023 1000000000027 1000000000029 1000000000030 1000000000031
1000000000033 1000000000037 1000000000038 1000000000039 1000000000041
1000000000042 1000000000043 1000000000045 1000000000046 1000000000047
1000000000049 1000000000051 1000000000054 1000000000055 1000000000057
1000000000058 1000000000059 1000000000061 1000000000063 1000000000065
1000000000066 1000000000067 1000000000069 1000000000070 1000000000073
1000000000074 1000000000077 1000000000078 1000000000079 1000000000081
1000000000082 1000000000085 1000000000086 1000000000087 1000000000090
1000000000091 1000000000093 1000000000094 1000000000095 1000000000097
1000000000099 1000000000101 1000000000102 1000000000103 1000000000105
1000000000106 1000000000109 1000000000111 1000000000113 1000000000114
1000000000115 1000000000117 1000000000118 1000000000119 1000000000121
1000000000122 1000000000123 1000000000126 1000000000127 1000000000129
1000000000130 1000000000133 1000000000135 1000000000137 1000000000138
1000000000139 1000000000141 1000000000142 1000000000145

The number of square─free numbers between 1 and 10^2 (inclusive) is: 61
The number of square─free numbers between 1 and 10^3 (inclusive) is: 608
The number of square─free numbers between 1 and 10^4 (inclusive) is: 6083
The number of square─free numbers between 1 and 10^5 (inclusive) is: 60794
The number of square─free numbers between 1 and 10^6 (inclusive) is: 607926
``````