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Challenges

# Determine whether an integer is square-free

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An integer is called square-free if it is not a multiple of a perfect square other than 1. For example, 42 is square-free, but 44 is not because it is a multiple of the perfect square 4 = 2².

Your task is to write a program or function that takes a positive integer, and returns a truthy value if the integer is square-free and a falsey value otherwise.

This is code golf, the shortest code wins.

The square-free numbers are OEIS sequence A005117 (thanks to Razetime for pointing this out).

Some test cases:

  1  true
2  true
3  true
4  false
5  true
6  true
7  true
8  false
9  false
10  true
12  false
14  true
16  false
18  false
20  false
30  true
40  false
50  false
100  false
110  true
111  true

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# Vyxal, 4 bytes

K∆²a


Try it Online!

Outputs 0 for square-free, 1 for not.

K    # Factors
∆²  # Is perfect square
a # Any true?

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# Python3, 39 bytes

lambda n:all(n%i**2for i in range(2,n))


Try it online!

Makes a list comprehension from the numbers 2 through n of the remainder of the square, and then checks whether the list contains 0

-8 bytes thanks to celtschk‭
-5 bytes thanks to Moshi

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not 0 in X => all X (2 comments)
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# Scala, 41 bytes

x=>2 to x forall(d=>x%d+math.sqrt(d)%1>0)


Try it online!

Pretty straightforward

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# Ruby, 27 bytes

->n{(2..n).all?{n%_1**2>0}}


Try it online

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# J, 17 bytes

{{*./y|~*:2+i.y}}


Try it online!

A direct definition closest to Razetime's infuriatingly good train solution. Outputs a non-zero number for true and 0 for false.

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# Myby, 12 5 bytes

primf=primfd

primf        : prime factors
=       : equals
primfd : unique prime factors


Evaluated as a monadic fork in J (f y) g (h y).

The test cases (retested) can be viewed here and were generated using this ruby script.

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# J, 7 char

*./~:q:


How it works:

'q:' produces prime factors of number on right

'~:' replaces the first instance of unique numbers with a 1, the rest 0

'*./' tests for all ones

Sample runs:

*./~:q: 30

1

*./~:q: 40

0

*./~:q: 50

0

*./~:q: 100

0

*./~:q: 110

1

*./~:q: 111

1

Alternative 10 char solution:

(#=#@~.)q:


How it works:

'q:' of the number on the right produces its prime factors

'#' counts how many there are

'~.' eliminates repeats.

if the counts of factors before and after removing repeats are equal, then the number is square-free.

Sample runs:

(#@q:=#@~.@q:) 30

1

(#@q:=#@~.@q:) 40

0

(#@q:=#@~.@q:) 50

0

(#@q:=#@~.@q:) 100

0

(#@q:=#@~.@q:) 110

1

(#@q:=#@~.@q:) 111

1

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# MATL, 6 bytes

YftuX=


Try it online!

Same method as Moshi.

YftuX=
Yf      - factor with implicit input
t    - duplicate
u   - unique
X= - isequal

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# Japt, 5 bytes

k
eUâ


Try it

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# BQN, 13 bytesSBCS

∧´0≠⊢|˜·×˜2+↕


Run online!

A train submission. It's 2 bytes shorter than the lambda version {∧´0≠𝕩|˜×˜2+↕𝕩}, due to omitting curly braces.

The idea is similar to ruby:

• range 2..n+1
• square it, then modulo n
• are all remainders ≠ 0?
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# Japt, 6 bytes

-1 byte thanks to Shaggy!

k ä¦ e


Try it

First time doing Japt so this is probably pretty bad. Just factorizes and checks that there are no duplicate factors.

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You missed a shortcut (2 comments)
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# JavaScript, 31 bytes

Outputs 0 for falsey and a non-zero value for truthy. If the 2 values must be consistent then replace the last * with && to output true instead.

n=>(g=d=>d++>n||n%d**2*g(d))(1)


Try it online!

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# Factor, 108 bytes

USING: assocs kernel math math.primes.factors ;
IN: s
: ? ( n -- ? ) group-factors [ nip 2 < ] assoc-all? ;

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