Möbius function

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Task
Möbius function
You are encouraged to solve this task according to the task description, using any language you may know.

The classical Möbius function: μ(n) is an important multiplicative function in number theory and combinatorics.

There are several ways to implement a Möbius function.

A fairly straightforward method is to find the prime factors of a positive integer n, then define μ(n) based on the sum of the primitive factors. It has the values {−1, 0, 1} depending on the factorization of n:

  • μ(1) is defined to be 1.
  • μ(n) = 1 if n is a square-free positive integer with an even number of prime factors.
  • μ(n) = −1 if n is a square-free positive integer with an odd number of prime factors.
  • μ(n) = 0 if n has a squared prime factor.


Task
  • Write a routine (function, procedure, whatever) μ(n) to find the Möbius number for a positive integer n.
  • Use that routine to find and display here, on this page, at least the first 99 terms in a grid layout. (Not just one long line or column of numbers.)


See also


Related Tasks



ALGOL 68

Translation of: C
BEGIN
    # show the first 199 values of the moebius function                 #
    INT sq root = 1 000;
    INT mu max  = sq root * sq root;
    [ 1 : mu max ]INT mu;
    FOR i FROM LWB mu TO UPB mu DO mu[ i ] := 1 OD;
    FOR i FROM 2 TO sq root DO
        IF mu[ i ] = 1 THEN
            # for each factor found, swap + and -                       #
            FOR j FROM i     BY i     TO UPB mu DO mu[ j ] *:= -i OD;
            FOR j FROM i * i BY i * i TO UPB mu DO mu[ j ]  :=  0 OD
        FI
    OD;
    FOR i FROM 2 TO UPB mu DO
        IF   mu[ i ] =  i THEN mu[ i ] :=  1
        ELIF mu[ i ] = -i THEN mu[ i ] := -1
        ELIF mu[ i ] <  0 THEN mu[ i ] :=  1
        ELIF mu[ i ] >  0 THEN mu[ i ] := -1
      # ELSE mu[ i ] =  0 so no change #
        FI
    OD;
    print( ( "First 199 terms of the möbius function are as follows:", newline, "    " ) );
    FOR i TO 199 DO
        print( ( whole( mu[ i ], -4 ) ) );
        IF ( i + 1 ) MOD 20 = 0 THEN print( ( newline ) ) FI
    OD
END
Output:
First 199 terms of the möbius function are as follows:
       1  -1  -1   0  -1   1  -1   0   0   1  -1   0  -1   1   1   0  -1   0  -1
   0   1   1  -1   0   0   1   0   0  -1  -1  -1   0   1   1   1   0  -1   1   1
   0  -1  -1  -1   0   0   1  -1   0   0   0   1   0  -1   0   1   0   1   1  -1
   0  -1   1   0   0   1  -1  -1   0   1  -1  -1   0  -1   1   0   0   1  -1  -1
   0   0   1  -1   0   1   1   1   0  -1   0   1   0   1   1   1   0  -1   0   0
   0  -1  -1  -1   0  -1   1  -1   0  -1  -1   1   0  -1  -1   1   0   0   1   1
   0   0   1   1   0   0   0  -1   0   1  -1  -1   0   1   1   0   0  -1  -1  -1
   0   1   1   1   0   1   1   0   0  -1   0  -1   0   0  -1   1   0  -1   1   1
   0   1   0  -1   0  -1   1  -1   0   0  -1   0   0  -1  -1   0   0   1   1  -1
   0  -1  -1   1   0   1  -1   1   0   0  -1  -1   0  -1   1  -1   0  -1   0  -1

Arturo

mobius: function [n][
    if n=0 -> return ""
    if n=1 -> return 1
    f: factors.prime n

    if f <> unique f -> return 0
    if? odd? size f -> return neg 1
    else -> return 1
]

loop split.every:20 map 0..199 => mobius 'a ->
    print map a => [pad to :string & 3]
Output:
      1  -1  -1   0  -1   1  -1   0   0   1  -1   0  -1   1   1   0  -1   0  -1 
  0   1   1  -1   0   0   1   0   0  -1  -1  -1   0   1   1   1   0  -1   1   1 
  0  -1  -1  -1   0   0   1  -1   0   0   0   1   0  -1   0   1   0   1   1  -1 
  0  -1   1   0   0   1  -1  -1   0   1  -1  -1   0  -1   1   0   0   1  -1  -1 
  0   0   1  -1   0   1   1   1   0  -1   0   1   0   1   1   1   0  -1   0   0 
  0  -1  -1  -1   0  -1   1  -1   0  -1  -1   1   0  -1  -1   1   0   0   1   1 
  0   0   1   1   0   0   0  -1   0   1  -1  -1   0   1   1   0   0  -1  -1  -1 
  0   1   1   1   0   1   1   0   0  -1   0  -1   0   0  -1   1   0  -1   1   1 
  0   1   0  -1   0  -1   1  -1   0   0  -1   0   0  -1  -1   0   0   1   1  -1 
  0  -1  -1   1   0   1  -1   1   0   0  -1  -1   0  -1   1  -1   0  -1   0  -1

AutoHotkey

loop 100
    result .= SubStr("  " Möbius(A_Index), -1) . (Mod(A_Index, 10) ? "  " : "`n")
MsgBox, 262144, , % result
return

Möbius(n){
    if n=1
        return 1
    x := prime_factors(n)
    c := x.Count()
    sq := []
    for i, v in x
        if sq[v]
            return 0
        else
            sq[v] := 1
    return (c/2 = floor(c/2)) ? 1 : -1
}

prime_factors(n) {
    if (n <= 3)
        return [n]
    ans := [], done := false
    while !done {
        if !Mod(n, 2)
            ans.push(2), n /= 2
        else if !Mod(n, 3)
            ans.push(3), n /= 3
        else if (n = 1)
            return ans
        else {
            sr := sqrt(n), done := true, i := 6
            while (i <= sr+6) {
                if !Mod(n, i-1) { ; is n divisible by i-1?
                    ans.push(i-1), n /= i-1, done := false
                    break
                }
                if !Mod(n, i+1) { ; is n divisible by i+1?
                    ans.push(i+1), n /= i+1, done := false
                    break
                }
                i += 6
    }}}
    ans.push(Format("{:d}", n))
    return ans
}
Output:
 1  -1  -1   0  -1   1  -1   0   0   1
-1   0  -1   1   1   0  -1   0  -1   0
 1   1  -1   0   0   1   0   0  -1  -1
-1   0   1   1   1   0  -1   1   1   0
-1  -1  -1   0   0   1  -1   0   0   0
 1   0  -1   0   1   0   1   1  -1   0
-1   1   0   0   1  -1  -1   0   1  -1
-1   0  -1   1   0   0   1  -1  -1   0
 0   1  -1   0   1   1   1   0  -1   0
 1   0   1   1   1   0  -1   0   0   0

AWK

# syntax: GAWK -f MOBIUS_FUNCTION.AWK
# converted from Java
BEGIN {
    printf("first 199 terms of the mobius sequence:\n   ")
    for (n=1; n<200; n++) {
      printf("%3d",mobius(n))
      if ((n+1) % 20 == 0) {
        printf("\n")
      }
    }
    exit(0)
}
function mobius(n,  i,j,mu_max) {
    if (n in MU) {
      return(MU[n])
    }
    mu_max = 1000000
    for (i=0; i<mu_max; i++) { # populate array
      MU[i] = 1
    }
    for (i=2; i<=int(sqrt(mu_max)); i++ ) {
      if (MU[i] == 1) {
        for (j=i; j<=mu_max; j+=i) { # for each factor found, swap + and -
          MU[j] *= -i
        }
        for (j=i*i; j<=mu_max; j+=i*i) { # square factor = 0
          MU[j] = 0
        }
      }
    }
    for (i=2; i<=mu_max; i++) {
      if (MU[i] == i) {
        MU[i] = 1
      }
      else if (MU[i] == -i) {
        MU[i] = -1
      }
      else if (MU[i] < 0) {
        MU[i] = 1
      }
      else if (MU[i] > 0) {
        MU[i] = -1
      }
    }
    return(MU[n])
}
Output:
first 199 terms of the mobius sequence:
     1 -1 -1  0 -1  1 -1  0  0  1 -1  0 -1  1  1  0 -1  0 -1
  0  1  1 -1  0  0  1  0  0 -1 -1 -1  0  1  1  1  0 -1  1  1
  0 -1 -1 -1  0  0  1 -1  0  0  0  1  0 -1  0  1  0  1  1 -1
  0 -1  1  0  0  1 -1 -1  0  1 -1 -1  0 -1  1  0  0  1 -1 -1
  0  0  1 -1  0  1  1  1  0 -1  0  1  0  1  1  1  0 -1  0  0
  0 -1 -1 -1  0 -1  1 -1  0 -1 -1  1  0 -1 -1  1  0  0  1  1
  0  0  1  1  0  0  0 -1  0  1 -1 -1  0  1  1  0  0 -1 -1 -1
  0  1  1  1  0  1  1  0  0 -1  0 -1  0  0 -1  1  0 -1  1  1
  0  1  0 -1  0 -1  1 -1  0  0 -1  0  0 -1 -1  0  0  1  1 -1
  0 -1 -1  1  0  1 -1  1  0  0 -1 -1  0 -1  1 -1  0 -1  0 -1


BASIC256

Translation of: FreeBASIC
function mobius(n)
	if n = 1 then return 1
	for d = 2 to int(sqr(n))
		if n mod d = 0 then
			if n mod (d*d) = 0 then return 0
			return -mobius(n/d)
		end if
	next d
	return -1
end function

outstr$ = " .   "
for i = 1 to 200
	if mobius(i) >= 0 then outstr$ += " "
	outstr$ += string(mobius(i)) + "   "
	if i mod 10 = 9 then
		print outstr$
		outstr$ = ""
	end if
next i
end
Output:
Igual que la entrada de FreeBASIC.


C

Translation of: Java
#include <math.h>
#include <stdio.h>
#include <stdlib.h>
#include <string.h>

int main() {
    const int MU_MAX = 1000000;
    int i, j;
    int *mu;
    int sqroot;

    sqroot = (int)sqrt(MU_MAX);

    mu = malloc((MU_MAX + 1) * sizeof(int));

    for (i = 0; i < MU_MAX;i++) {
        mu[i] = 1;
    }

    for (i = 2; i <= sqroot; i++) {
        if (mu[i] == 1) {
            // for each factor found, swap + and -
            for (j = i; j <= MU_MAX; j += i) {
                mu[j] *= -i;
            }
            // square factor = 0
            for (j = i * i; j <= MU_MAX; j += i * i) {
                mu[j] = 0;
            }
        }
    }

    for (i = 2; i <= MU_MAX; i++) {
        if (mu[i] == i) {
            mu[i] = 1;
        } else if (mu[i] == -i) {
            mu[i] = -1;
        } else if (mu[i] < 0) {
            mu[i] = 1;
        } else if (mu[i] > 0) {
            mu[i] = -1;
        }
    }

    printf("First 199 terms of the möbius function are as follows:\n    ");
    for (i = 1; i < 200; i++) {
        printf("%2d  ", mu[i]);
        if ((i + 1) % 20 == 0) {
            printf("\n");
        }
    }

    free(mu);
    return 0;
}
Output:
First 199 terms of the m÷bius function are as follows:
     1  -1  -1   0  -1   1  -1   0   0   1  -1   0  -1   1   1   0  -1   0  -1
 0   1   1  -1   0   0   1   0   0  -1  -1  -1   0   1   1   1   0  -1   1   1
 0  -1  -1  -1   0   0   1  -1   0   0   0   1   0  -1   0   1   0   1   1  -1
 0  -1   1   0   0   1  -1  -1   0   1  -1  -1   0  -1   1   0   0   1  -1  -1
 0   0   1  -1   0   1   1   1   0  -1   0   1   0   1   1   1   0  -1   0   0
 0  -1  -1  -1   0  -1   1  -1   0  -1  -1   1   0  -1  -1   1   0   0   1   1
 0   0   1   1   0   0   0  -1   0   1  -1  -1   0   1   1   0   0  -1  -1  -1
 0   1   1   1   0   1   1   0   0  -1   0  -1   0   0  -1   1   0  -1   1   1
 0   1   0  -1   0  -1   1  -1   0   0  -1   0   0  -1  -1   0   0   1   1  -1
 0  -1  -1   1   0   1  -1   1   0   0  -1  -1   0  -1   1  -1   0  -1   0  -1

C++

Translation of: Java
#include <iomanip>
#include <iostream>
#include <vector>

constexpr int MU_MAX = 1'000'000;
std::vector<int> MU;

int mobiusFunction(int n) {
    if (!MU.empty()) {
        return MU[n];
    }

    // Populate array
    MU.resize(MU_MAX + 1, 1);
    int root = sqrt(MU_MAX);

    for (int i = 2; i <= root; i++) {
        if (MU[i] == 1) {
            // for each factor found, swap + and -
            for (int j = i; j <= MU_MAX; j += i) {
                MU[j] *= -i;
            }
            // square factor = 0
            for (int j = i * i; j <= MU_MAX; j += i * i) {
                MU[j] = 0;
            }
        }
    }

    for (int i = 2; i <= MU_MAX; i++) {
        if (MU[i] == i) {
            MU[i] = 1;
        } else if (MU[i] == -i) {
            MU[i] = -1;
        } else if (MU[i] < 0) {
            MU[i] = 1;
        } else if (MU[i] > 0) {
            MU[i] = -1;
        }
    }

    return MU[n];
}

int main() {
    std::cout << "First 199 terms of the möbius function are as follows:\n    ";
    for (int n = 1; n < 200; n++) {
        std::cout << std::setw(2) << mobiusFunction(n) << "  ";
        if ((n + 1) % 20 == 0) {
            std::cout << '\n';
        }
    }

    return 0;
}
Output:
First 199 terms of the m÷bius function are as follows:
     1  -1  -1   0  -1   1  -1   0   0   1  -1   0  -1   1   1   0  -1   0  -1
 0   1   1  -1   0   0   1   0   0  -1  -1  -1   0   1   1   1   0  -1   1   1
 0  -1  -1  -1   0   0   1  -1   0   0   0   1   0  -1   0   1   0   1   1  -1
 0  -1   1   0   0   1  -1  -1   0   1  -1  -1   0  -1   1   0   0   1  -1  -1
 0   0   1  -1   0   1   1   1   0  -1   0   1   0   1   1   1   0  -1   0   0
 0  -1  -1  -1   0  -1   1  -1   0  -1  -1   1   0  -1  -1   1   0   0   1   1
 0   0   1   1   0   0   0  -1   0   1  -1  -1   0   1   1   0   0  -1  -1  -1
 0   1   1   1   0   1   1   0   0  -1   0  -1   0   0  -1   1   0  -1   1   1
 0   1   0  -1   0  -1   1  -1   0   0  -1   0   0  -1  -1   0   0   1   1  -1
 0  -1  -1   1   0   1  -1   1   0   0  -1  -1   0  -1   1  -1   0  -1   0  -1

D

Translation of: C++
import std.math;
import std.stdio;

immutable MU_MAX = 1_000_000;

int mobiusFunction(int n) {
    static initialized = false;
    static int[MU_MAX + 1] MU;

    if (initialized) {
        return MU[n];
    }

    // populate array
    MU[] = 1;
    int root = cast(int) sqrt(cast(real) MU_MAX);

    for (int i = 2; i <= root; i++) {
        if (MU[i] == 1) {
            // for each factor found, swap + and -
            for (int j = i; j <= MU_MAX; j += i) {
                MU[j] *= -i;
            }
            // square factor = 0
            for (int j = i * i; j <= MU_MAX; j += i * i) {
                MU[j] = 0;
            }
        }
    }

    for (int i = 2; i <= MU_MAX; i++) {
        if (MU[i] == i) {
            MU[i] = 1;
        } else if (MU[i] == -i) {
            MU[i] = -1;
        } else if (MU[i] < 0) {
            MU[i] = 1;
        } else if (MU[i] > 0) {
            MU[i] = -1;
        }
    }

    initialized = true;
    return MU[n];
}

void main() {
    writeln("First 199 terms of the möbius function are as follows:");
    write("    ");
    for (int n = 1; n < 200; n++) {
        writef("%2d  ", mobiusFunction(n));
        if ((n + 1) % 20 == 0) {
            writeln;
        }
    }
}
Output:
First 199 terms of the m├╢bius function are as follows:
     1  -1  -1   0  -1   1  -1   0   0   1  -1   0  -1   1   1   0  -1   0  -1
 0   1   1  -1   0   0   1   0   0  -1  -1  -1   0   1   1   1   0  -1   1   1
 0  -1  -1  -1   0   0   1  -1   0   0   0   1   0  -1   0   1   0   1   1  -1
 0  -1   1   0   0   1  -1  -1   0   1  -1  -1   0  -1   1   0   0   1  -1  -1
 0   0   1  -1   0   1   1   1   0  -1   0   1   0   1   1   1   0  -1   0   0
 0  -1  -1  -1   0  -1   1  -1   0  -1  -1   1   0  -1  -1   1   0   0   1   1
 0   0   1   1   0   0   0  -1   0   1  -1  -1   0   1   1   0   0  -1  -1  -1
 0   1   1   1   0   1   1   0   0  -1   0  -1   0   0  -1   1   0  -1   1   1
 0   1   0  -1   0  -1   1  -1   0   0  -1   0   0  -1  -1   0   0   1   1  -1
 0  -1  -1   1   0   1  -1   1   0   0  -1  -1   0  -1   1  -1   0  -1   0  -1

F#

This task uses Extensible Prime Generator (F#)

// Möbius function. Nigel Galloway: January 31st., 2021
let fN g=let n=primes32()
         let rec fN i g e l=match (l/g,l%g,e) with (1,0,false)->i
                                                  |(n,0,false)->fN (0-i) g true n
                                                  |(_,0,true) ->0
                                                  |_          ->fN i (Seq.head n) false l
         fN -1 (Seq.head n) false g
let mobius=seq{yield 1; yield! Seq.initInfinite((+)2>>fN)}
mobius|>Seq.take 500|>Seq.chunkBySize 25|>Seq.iter(fun n->Array.iter(printf "%3d") n;printfn "")
Output:
  1 -1 -1  0 -1  1 -1  0  0  1 -1  0 -1  1  1  0 -1  0 -1  0  1  1 -1  0  0
  1  0  0 -1 -1 -1  0  1  1  1  0 -1  1  1  0 -1 -1 -1  0  0  1 -1  0  0  0
  1  0 -1  0  1  0  1  1 -1  0 -1  1  0  0  1 -1 -1  0  1 -1 -1  0 -1  1  0
  0  1 -1 -1  0  0  1 -1  0  1  1  1  0 -1  0  1  0  1  1  1  0 -1  0  0  0
 -1 -1 -1  0 -1  1 -1  0 -1 -1  1  0 -1 -1  1  0  0  1  1  0  0  1  1  0  0
  0 -1  0  1 -1 -1  0  1  1  0  0 -1 -1 -1  0  1  1  1  0  1  1  0  0 -1  0
 -1  0  0 -1  1  0 -1  1  1  0  1  0 -1  0 -1  1 -1  0  0 -1  0  0 -1 -1  0
  0  1  1 -1  0 -1 -1  1  0  1 -1  1  0  0 -1 -1  0 -1  1 -1  0 -1  0 -1  0
  1  1  1  0  1  1  0  0  1  1 -1  0  1  1  1  0  1  1  1  0  1 -1 -1  0  0
  1 -1  0 -1 -1 -1  0 -1  0  1  0  1 -1 -1  0 -1  0  0  0  0 -1  1  0  1  0
 -1  0  1  1 -1  0 -1 -1  1  0  0  1 -1  0  1 -1  1  0 -1  0 -1  0 -1  1  0
  0 -1  1  0  0 -1 -1 -1  0 -1 -1  1  0  0 -1  1  0 -1  0  1  0  0  1  1  0
  1  1  1  0  1  0 -1  0  1 -1 -1  0 -1  1  0  0 -1 -1  1  0  1 -1  1  0  0
  1  1  0  1  1 -1  0  0  1  1  0 -1  0  1  0  1  0  0  0 -1  1 -1  0 -1  0
  0  0 -1 -1  1  0 -1  1 -1  0  0  1  0  0  1 -1 -1  0  0 -1  1  0 -1 -1  0
  0  1  0 -1  0  1  1 -1  0 -1  1  0  0 -1  1  1  0  1  1  1  0 -1  1 -1  0
 -1 -1  1  0  0 -1  1  0 -1 -1  1  0  1  0  1  0  1 -1 -1  0 -1  1  0  0  0
 -1  1  0 -1 -1 -1  0 -1 -1 -1  0  1 -1 -1  0  0 -1 -1  0  1  1  1  0 -1  0
  1  0  1  1 -1  0 -1  1  0  0 -1  1 -1  0 -1  1 -1  0  1 -1  1  0  1 -1  0
  0  0  1 -1  0  1  1 -1  0  1  0 -1  0  1  0 -1  0  1 -1  0  0  1 -1 -1  0

Factor

The mobius word exists in the math.extras vocabulary. See the implementation here.

Works with: Factor version 0.99 2020-01-23
USING: formatting grouping io math.extras math.ranges sequences ;

"First 199 terms of the Möbius sequence:" print
199 [1,b] [ mobius ] map " " prefix 20 group
[ [ "%3s" printf ] each nl ] each
Output:
First 199 terms of the Möbius sequence:
     1 -1 -1  0 -1  1 -1  0  0  1 -1  0 -1  1  1  0 -1  0 -1
  0  1  1 -1  0  0  1  0  0 -1 -1 -1  0  1  1  1  0 -1  1  1
  0 -1 -1 -1  0  0  1 -1  0  0  0  1  0 -1  0  1  0  1  1 -1
  0 -1  1  0  0  1 -1 -1  0  1 -1 -1  0 -1  1  0  0  1 -1 -1
  0  0  1 -1  0  1  1  1  0 -1  0  1  0  1  1  1  0 -1  0  0
  0 -1 -1 -1  0 -1  1 -1  0 -1 -1  1  0 -1 -1  1  0  0  1  1
  0  0  1  1  0  0  0 -1  0  1 -1 -1  0  1  1  0  0 -1 -1 -1
  0  1  1  1  0  1  1  0  0 -1  0 -1  0  0 -1  1  0 -1  1  1
  0  1  0 -1  0 -1  1 -1  0  0 -1  0  0 -1 -1  0  0  1  1 -1
  0 -1 -1  1  0  1 -1  1  0  0 -1 -1  0 -1  1 -1  0 -1  0 -1

Fortran

Translation of: C
program moebius
    use iso_fortran_env, only: output_unit

    integer, parameter          :: mu_max=1000000, line_break=20
    integer, parameter          :: sqroot=int(sqrt(real(mu_max)))
    integer                     :: i, j
    integer, dimension(mu_max)  :: mu

    mu = 1

    do i = 2, sqroot
        if (mu(i) == 1) then
            do j = i, mu_max, i
                mu(j) = mu(j) * (-i)
            end do

            do j = i**2, mu_max, i**2
                mu(j) = 0
            end do
        end if
    end do

    do i = 2, mu_max
        if (mu(i) == i) then
            mu(i) = 1
        else if (mu(i) == -i) then
            mu(i) = -1
        else if (mu(i) < 0) then
            mu(i) = 1
        else if (mu(i) > 0) then
            mu(i) = -1
        end if
    end do

    write(output_unit,*) "The first 199 terms of the Möbius sequence are:"
    write(output_unit,'(3x)', advance="no") ! Alignment of first number
    do i = 1, 199
        write(output_unit,'(I2,x)', advance="no") mu(i)
        if (modulo(i+1, line_break) == 0) write(output_unit,*)
    end do
end program moebius
Output:
 The first 199 terms of the Möbius sequence are:
    1 -1 -1  0 -1  1 -1  0  0  1 -1  0 -1  1  1  0 -1  0 -1 
 0  1  1 -1  0  0  1  0  0 -1 -1 -1  0  1  1  1  0 -1  1  1 
 0 -1 -1 -1  0  0  1 -1  0  0  0  1  0 -1  0  1  0  1  1 -1 
 0 -1  1  0  0  1 -1 -1  0  1 -1 -1  0 -1  1  0  0  1 -1 -1 
 0  0  1 -1  0  1  1  1  0 -1  0  1  0  1  1  1  0 -1  0  0 
 0 -1 -1 -1  0 -1  1 -1  0 -1 -1  1  0 -1 -1  1  0  0  1  1 
 0  0  1  1  0  0  0 -1  0  1 -1 -1  0  1  1  0  0 -1 -1 -1 
 0  1  1  1  0  1  1  0  0 -1  0 -1  0  0 -1  1  0 -1  1  1 
 0  1  0 -1  0 -1  1 -1  0  0 -1  0  0 -1 -1  0  0  1  1 -1 
 0 -1 -1  1  0  1 -1  1  0  0 -1 -1  0 -1  1 -1  0 -1  0 -1 

FreeBASIC

function mobius( n as uinteger ) as integer
    if n = 1 then return 1
    for d as uinteger = 2 to int(sqr(n))
        if n mod d = 0 then 
            if n mod (d*d) = 0 then return 0
            return -mobius(n/d)
        end if
    next d
    return -1
end function

dim as string outstr = " .     "
for i as uinteger = 1 to 200
    if mobius(i)>=0 then outstr += " "
    outstr += str(mobius(i))+"     "
    if i mod 10 = 9 then 
        print outstr
        outstr = ""
    end if
next i
Output:
 .      1     -1     -1      0     -1      1     -1      0      0     
 1     -1      0     -1      1      1      0     -1      0     -1     
 0      1      1     -1      0      0      1      0      0     -1     
-1     -1      0      1      1      1      0     -1      1      1     
 0     -1     -1     -1      0      0      1     -1      0      0     
 0      1      0     -1      0      1      0      1      1     -1     
 0     -1      1      0      0      1     -1     -1      0      1     
-1     -1      0     -1      1      0      0      1     -1     -1     
 0      0      1     -1      0      1      1      1      0     -1     
 0      1      0      1      1      1      0     -1      0      0     
 0     -1     -1     -1      0     -1      1     -1      0     -1     
-1      1      0     -1     -1      1      0      0      1      1     
 0      0      1      1      0      0      0     -1      0      1     
-1     -1      0      1      1      0      0     -1     -1     -1     
 0      1      1      1      0      1      1      0      0     -1     
 0     -1      0      0     -1      1      0     -1      1      1     
 0      1      0     -1      0     -1      1     -1      0      0     
-1      0      0     -1     -1      0      0      1      1     -1     
 0     -1     -1      1      0      1     -1      1      0      0     
-1     -1      0     -1      1     -1      0     -1      0     -1


FutureBasic

local fn IsPrime( n as long ) as BOOL
  BOOL result = YES
  long i
  
  if ( n < 2 ) then result = NO : exit fn
  
  for i = 2 to n + 1
    if ( i * i <= n ) and ( n mod i == 0 )
      result = NO : exit fn
    end if
  next
end fn = result

local fn Mobius( n as long ) as long
  long i, p = 0, result = 0
  
  if ( n == 1 ) then result = 1 : exit fn
  
  for i = 1 to n + 1
    if ( n mod i == 0 ) and ( fn IsPrime( i ) == YES )
      if ( n mod ( i * i ) == 0 )
        result = 0 : exit fn
      else
        p++
      end if
    end if
  next
  
  if( p mod 2 != 0 )
    result = -1
  else
    result = 1
  end if
end fn = result

window 1, @"Möbius function", (0,0,600,300)

printf @"First 100 terms of Mobius sequence:"

long i
for i = 1 to 100
  printf @"%2ld\t", fn Mobius(i)
  if ( i mod 20 == 0 ) then print
next

HandleEvents
Output:
First 100 terms of Mobius sequence:
   1  -1  -1   0  -1   1  -1   0   0   1  -1   0  -1   1   1   0  -1   0  -1   0
   1   1  -1   0   0   1   0   0  -1  -1  -1   0   1   1   1   0  -1   1   1   0
  -1  -1  -1   0   0   1  -1   0   0   0   1   0  -1   0   1   0   1   1  -1   0
  -1   1   0   0   1  -1  -1   0   1  -1  -1   0  -1   1   0   0   1  -1  -1   0
   0   1  -1   0   1   1   1   0  -1   0   1   0   1   1   1   0  -1   0   0   0

Go

package main

import "fmt"

func möbius(to int) []int {
    if to < 1 {
        to = 1
    }
    mobs := make([]int, to+1) // all zero by default
    primes := []int{2}
    for i := 1; i <= to; i++ {
        j := i
        cp := 0      // counts prime factors
        spf := false // true if there is a square prime factor
        for _, p := range primes {
            if p > j {
                break
            }
            if j%p == 0 {
                j /= p
                cp++
            }
            if j%p == 0 {
                spf = true
                break
            }
        }
        if cp == 0 && i > 2 {
            cp = 1
            primes = append(primes, i)
        }
        if !spf {
            if cp%2 == 0 {
                mobs[i] = 1
            } else {
                mobs[i] = -1
            }
        }
    }
    return mobs
}

func main() {
    mobs := möbius(199)
    fmt.Println("Möbius sequence - First 199 terms:")
    for i := 0; i < 200; i++ {
        if i == 0 {
            fmt.Print("    ")
            continue
        }
        if i%20 == 0 {
            fmt.Println()
        }
        fmt.Printf("  % d", mobs[i])
    }
}
Output:
Möbius sequence - First 199 terms:
       1  -1  -1   0  -1   1  -1   0   0   1  -1   0  -1   1   1   0  -1   0  -1
   0   1   1  -1   0   0   1   0   0  -1  -1  -1   0   1   1   1   0  -1   1   1
   0  -1  -1  -1   0   0   1  -1   0   0   0   1   0  -1   0   1   0   1   1  -1
   0  -1   1   0   0   1  -1  -1   0   1  -1  -1   0  -1   1   0   0   1  -1  -1
   0   0   1  -1   0   1   1   1   0  -1   0   1   0   1   1   1   0  -1   0   0
   0  -1  -1  -1   0  -1   1  -1   0  -1  -1   1   0  -1  -1   1   0   0   1   1
   0   0   1   1   0   0   0  -1   0   1  -1  -1   0   1   1   0   0  -1  -1  -1
   0   1   1   1   0   1   1   0   0  -1   0  -1   0   0  -1   1   0  -1   1   1
   0   1   0  -1   0  -1   1  -1   0   0  -1   0   0  -1  -1   0   0   1   1  -1
   0  -1  -1   1   0   1  -1   1   0   0  -1  -1   0  -1   1  -1   0  -1   0  -1

GW-BASIC

10 FOR T = 0 TO 9
20 FOR U = 1 TO 10
30 N = 10*T + U
40 GOSUB 100
50 PRINT USING "##  ";M;
60 NEXT U
70 PRINT
80 NEXT T
90 END
100 IF N = 1 THEN M = 1 : RETURN
110 M = 1 : F = 2
120 IF N MOD (F*F) = 0 THEN M = 0 : RETURN
130 IF N MOD F = 0 THEN GOSUB 170
140 F = F + 1
150 IF F <= N THEN GOTO 120
160 RETURN
170 M = -M
180 N = N/F
190 RETURN

Haskell

import Data.List (intercalate)
import Data.List.Split (chunksOf)
import Data.Vector.Unboxed (toList)
import Math.NumberTheory.ArithmeticFunctions.Moebius (Moebius(..),
                                                      sieveBlockMoebius)
import System.Environment (getArgs, getProgName)
import System.IO (hPutStrLn, stderr)
import Text.Read (readMaybe)

-- Calculate the Möbius function, μ(n), for a sequence of values starting at 1.
moebiusBlock :: Word -> [Moebius]
moebiusBlock = toList . sieveBlockMoebius 1

showMoebiusBlock :: Word -> [Moebius] -> String
showMoebiusBlock cols = intercalate "\n" . map (concatMap showMoebius) .
                        chunksOf (fromIntegral cols)
  where showMoebius MoebiusN = " -1"
        showMoebius MoebiusZ = "  0"
        showMoebius MoebiusP = "  1"

main :: IO ()
main = do
  prog <- getProgName
  args <- map readMaybe <$> getArgs
  case args of
    [Just cols, Just n] ->
      putStrLn ("μ(n) for 1 ≤ n ≤ " ++ show n ++ ":\n") >>
      putStrLn (showMoebiusBlock cols $ moebiusBlock n)
    _ -> hPutStrLn stderr $ "Usage: " ++ prog ++ " num-columns maximum-number"
Output:
μ(n) for 1 ≤ n ≤ 200:

  1 -1 -1  0 -1  1 -1  0  0  1 -1  0 -1  1  1  0 -1  0 -1  0
  1  1 -1  0  0  1  0  0 -1 -1 -1  0  1  1  1  0 -1  1  1  0
 -1 -1 -1  0  0  1 -1  0  0  0  1  0 -1  0  1  0  1  1 -1  0
 -1  1  0  0  1 -1 -1  0  1 -1 -1  0 -1  1  0  0  1 -1 -1  0
  0  1 -1  0  1  1  1  0 -1  0  1  0  1  1  1  0 -1  0  0  0
 -1 -1 -1  0 -1  1 -1  0 -1 -1  1  0 -1 -1  1  0  0  1  1  0
  0  1  1  0  0  0 -1  0  1 -1 -1  0  1  1  0  0 -1 -1 -1  0
  1  1  1  0  1  1  0  0 -1  0 -1  0  0 -1  1  0 -1  1  1  0
  1  0 -1  0 -1  1 -1  0  0 -1  0  0 -1 -1  0  0  1  1 -1  0
 -1 -1  1  0  1 -1  1  0  0 -1 -1  0 -1  1 -1  0 -1  0 -1  0

J

Implementation:

mu=: ({{*/1-y>1}} * _1 ^ 2|+/)@q:~&_

Explanation: _ q: n gives the list of exponents of prime factors of n. (This is an empty list for the number 1, is 2 0 2 for the number 100 and is 3 1 1 for the number 120.)

In this context +/ is the sum of that list, 2 | +/ is 1 if the sum is odd and 0 if the sum is even. _1^0 is 1 and _1^1 is _1. And, {{*/1-y>1}} returns zero if any exponent is at least 2 in magnitude.

Task example:

   mu 1+i.10 20
 1 _1 _1 0 _1  1 _1 0  0  1 _1 0 _1  1  1 0 _1  0 _1 0
 1  1 _1 0  0  1  0 0 _1 _1 _1 0  1  1  1 0 _1  1  1 0
_1 _1 _1 0  0  1 _1 0  0  0  1 0 _1  0  1 0  1  1 _1 0
_1  1  0 0  1 _1 _1 0  1 _1 _1 0 _1  1  0 0  1 _1 _1 0
 0  1 _1 0  1  1  1 0 _1  0  1 0  1  1  1 0 _1  0  0 0
_1 _1 _1 0 _1  1 _1 0 _1 _1  1 0 _1 _1  1 0  0  1  1 0
 0  1  1 0  0  0 _1 0  1 _1 _1 0  1  1  0 0 _1 _1 _1 0
 1  1  1 0  1  1  0 0 _1  0 _1 0  0 _1  1 0 _1  1  1 0
 1  0 _1 0 _1  1 _1 0  0 _1  0 0 _1 _1  0 0  1  1 _1 0
_1 _1  1 0  1 _1  1 0  0 _1 _1 0 _1  1 _1 0 _1  0 _1 0

Java

public class MöbiusFunction {

    public static void main(String[] args) {
        System.out.printf("First 199 terms of the möbius function are as follows:%n    ");
        for ( int n = 1 ; n < 200 ; n++ ) {
            System.out.printf("%2d  ", möbiusFunction(n));
            if ( (n+1) % 20 == 0 ) {
                System.out.printf("%n");
            }
        }
    }
    
    private static int MU_MAX = 1_000_000;
    private static int[] MU = null;
    
    //  Compute mobius function via sieve
    private static int möbiusFunction(int n) {
        if ( MU != null ) {
            return MU[n];
        }
        
        //  Populate array
        MU = new int[MU_MAX+1];
        int sqrt = (int) Math.sqrt(MU_MAX);
        for ( int i = 0 ; i < MU_MAX ; i++ ) {
            MU[i] = 1;
        }
        
        for ( int i = 2 ; i <= sqrt ; i++ ) {
            if ( MU[i] == 1 ) {
                //  for each factor found, swap + and -
                for ( int j = i ; j <= MU_MAX ; j += i ) {
                    MU[j] *= -i;
                }
                //  square factor = 0
                for ( int j = i*i ; j <= MU_MAX ; j += i*i ) {
                    MU[j] = 0;
                }
            }
        }
        
        for ( int i = 2 ; i <= MU_MAX ; i++ ) {
            if ( MU[i] == i ) {
                MU[i] = 1;
            }
            else if ( MU[i] == -i ) {
                MU[i] = -1;
            }
            else if ( MU[i] < 0 ) {
                MU[i] = 1;               
            }
            else if ( MU[i] > 0 ) {
                MU[i] = -1;
            }
        }
        return MU[n];
    }

}
Output:
First 199 terms of the möbius function are as follows:
     1  -1  -1   0  -1   1  -1   0   0   1  -1   0  -1   1   1   0  -1   0  -1  
 0   1   1  -1   0   0   1   0   0  -1  -1  -1   0   1   1   1   0  -1   1   1  
 0  -1  -1  -1   0   0   1  -1   0   0   0   1   0  -1   0   1   0   1   1  -1  
 0  -1   1   0   0   1  -1  -1   0   1  -1  -1   0  -1   1   0   0   1  -1  -1  
 0   0   1  -1   0   1   1   1   0  -1   0   1   0   1   1   1   0  -1   0   0  
 0  -1  -1  -1   0  -1   1  -1   0  -1  -1   1   0  -1  -1   1   0   0   1   1  
 0   0   1   1   0   0   0  -1   0   1  -1  -1   0   1   1   0   0  -1  -1  -1  
 0   1   1   1   0   1   1   0   0  -1   0  -1   0   0  -1   1   0  -1   1   1  
 0   1   0  -1   0  -1   1  -1   0   0  -1   0   0  -1  -1   0   0   1   1  -1  
 0  -1  -1   1   0   1  -1   1   0   0  -1  -1   0  -1   1  -1   0  -1   0  -1  

jq

Works with: jq

Works with gojq, the Go implementation of jq

Using a Sieve

Adapted from C

# Input: a non-negative integer, $n
# Output: an array of size $n + 1 such that the nth-mobius number is .[$n]
# i.e. $n|mobius_array[-1]
# For example, the first mobius number could be evaluated by 1|mobius_array[-1].
def mobius_array:
  . as $n
  | ($n|sqrt) as $sqrt
  | reduce range(2; 1 + $sqrt) as $i ([range(0; $n + 1) | 1];
       if .[$i] == 1
       then # for each factor found, swap + and -
         reduce range($i; $n + 1; $i) as $j (.; .[$j] *= -$i) 
         | ($i*$i) as $isq #  square factor = 0
         | reduce range($isq; $n + 1; $isq) as $j (.; .[$j] = 0 )
       else .
       end )
  | reduce range(2; 1 + $n) as $i (.;
       if   .[$i] ==  $i then .[$i] = 1
       elif .[$i] == -$i then .[$i] = -1
       elif .[$i]  <   0 then .[$i] = 1
       elif .[$i]  >   0 then .[$i] = -1
       else .[$i] = 0                   # avoid "-0"
       end);

# For one-off computations:
def mu($n): $n | mobius_array[-1];

The Task

def nwise($n):
  def n: if length <= $n then . else .[0:$n] , (.[$n:] | n) end;
  n;

def task:
  def pp: if . >=0 then " \(.)" else tostring end;
  (199 | mobius_array) as $mu
  | "The first 199 Möbius numbers are:",
    (["  ", (range(1; 200) | $mu[.] | pp )]
     | nwise(20)
     | join(" ") ) ;

task
Output:
The first 199 Möbius numbers are:
    1 -1 -1  0 -1  1 -1  0  0  1 -1  0 -1  1  1  0 -1  0 -1
 0  1  1 -1  0  0  1  0  0 -1 -1 -1  0  1  1  1  0 -1  1  1
 0 -1 -1 -1  0  0  1 -1  0  0  0  1  0 -1  0  1  0  1  1 -1
 0 -1  1  0  0  1 -1 -1  0  1 -1 -1  0 -1  1  0  0  1 -1 -1
 0  0  1 -1  0  1  1  1  0 -1  0  1  0  1  1  1  0 -1  0  0
 0 -1 -1 -1  0 -1  1 -1  0 -1 -1  1  0 -1 -1  1  0  0  1  1
 0  0  1  1  0  0  0 -1  0  1 -1 -1  0  1  1  0  0 -1 -1 -1
 0  1  1  1  0  1  1  0  0 -1  0 -1  0  0 -1  1  0 -1  1  1
 0  1  0 -1  0 -1  1 -1  0  0 -1  0  0 -1 -1  0  0  1  1 -1
 0 -1 -1  1  0  1 -1  1  0  0 -1 -1  0 -1  1 -1  0 -1  0 -1

Prime Factors

Note that the following solution to the task at hand (computing a range of Mobius numbers is inefficient as it does not cache the primes array. Preliminaries

# relatively_prime(previous) tests whether the input integer is prime
# relative to the primes in the array "previous":
def relatively_prime(previous):
  . as $in
  | (previous|length) as $plen
  # state: [found, ix]
  | [false, 0]
  | until( .[0] or .[1] >= $plen;
           [ ($in % previous[.[1]]) == 0, .[1] + 1] )
  | .[0] | not ;

# Emit a stream in increasing order of all primes (from 2 onwards)
# that are less than or equal to mx:
def primes(mx):
  # The helper function, next, has arity 0 for tail recursion optimization;
  # it expects its input to be the array of previously found primes:
  def next:
     . as $previous
     | ($previous | .[length-1]) as $last
     | if ($last >= mx) then empty
       else ((2 + $last)
       | until( relatively_prime($previous) ; . + 2)) as $nextp
       | if $nextp <= mx
         then $nextp, (( $previous + [$nextp] ) | next)
	 else empty
         end
       end;
  if mx <= 1 then empty
  elif mx == 2 then 2
  else (2, 3, ([2,3] | next))
  end ;

# Return an array of the distinct prime factors of . in increasing order
def prime_factors:

  # Return an array of prime factors of . given that "primes"
  # is an array of relevant primes:
  def pf($primes):
    if . <= 1 then []
    else . as $in
    | if ($in | relatively_prime($primes)) then [$in]
      else reduce $primes[] as $p
             ([];
              if ($in % $p) != 0 then .
 	      else . + [$p] +  (($in / $p) | pf($primes))
	      end)
      end
      | unique
    end;
    
  if . <= 1 then []
  else . as $in
  | pf( [ primes( (1+$in) | sqrt | floor)  ] )
  end;

Mu

def isSquareFree:
  . as $n
  | 2
  | until ( (. * . > $n) or . == 0;
       if ($n % (.*.) == 0) then 0 # i.e. stop
       elif . > 2 then . + 2
       else . + 1
       end  )
  | . != 0;

def mu:
  . as $n
  | if . < 1 then "Argument to mu must be a positive integer" | error
    elif . == 1 then 1
    else if isSquareFree 
         then if ((prime_factors|length) % 2 == 0) then 1 
              else -1
              end
         else 0
         end
    end;

The Task

def nwise($n):
  def n: if length <= $n then . else .[0:$n] , (.[$n:] | n) end;
  n;

def task:
  def pp: if . >=0 then " \(.)" else tostring end;
  "The first 199 Möbius numbers are:",
  (["  ", (range(1; 200) | mu | pp )]
   | nwise(20)
   | join(" ") ) ;

task
Output:

As above.

Julia

using Primes

# modified from reinermartin's PR at https://github.com/JuliaMath/Primes.jl/pull/70/files
function moebius(n::Integer)
    @assert n > 0
    m(p, e) = p == 0 ? 0 : e == 1 ? -1 : 0
    reduce(*, m(p, e) for (p, e) in factor(n) if p  0; init=1)
end
μ(n) = moebius(n)

print("First 199 terms of the Möbius sequence:\n   ")
for n in 1:199
    print(lpad(μ(n), 3), n % 20 == 19 ? "\n" : "")
end
Output:
First 199 terms of the Möbius sequence:
     1 -1 -1  0 -1  1 -1  0  0  1 -1  0 -1  1  1  0 -1  0 -1
  0  1  1 -1  0  0  1  0  0 -1 -1 -1  0  1  1  1  0 -1  1  1
  0 -1 -1 -1  0  0  1 -1  0  0  0  1  0 -1  0  1  0  1  1 -1
  0 -1  1  0  0  1 -1 -1  0  1 -1 -1  0 -1  1  0  0  1 -1 -1
  0  0  1 -1  0  1  1  1  0 -1  0  1  0  1  1  1  0 -1  0  0
  0 -1 -1 -1  0 -1  1 -1  0 -1 -1  1  0 -1 -1  1  0  0  1  1
  0  0  1  1  0  0  0 -1  0  1 -1 -1  0  1  1  0  0 -1 -1 -1
  0  1  1  1  0  1  1  0  0 -1  0 -1  0  0 -1  1  0 -1  1  1
  0  1  0 -1  0 -1  1 -1  0  0 -1  0  0 -1 -1  0  0  1  1 -1
  0 -1 -1  1  0  1 -1  1  0  0 -1 -1  0 -1  1 -1  0 -1  0 -1

Kotlin

Translation of: Java
import kotlin.math.sqrt

fun main() {
    println("First 199 terms of the möbius function are as follows:")
    print("    ")
    for (n in 1..199) {
        print("%2d  ".format(mobiusFunction(n)))
        if ((n + 1) % 20 == 0) {
            println()
        }
    }
}

private const val MU_MAX = 1000000
private var MU: IntArray? = null

//  Compute mobius function via sieve
private fun mobiusFunction(n: Int): Int {
    if (MU != null) {
        return MU!![n]
    }

    //  Populate array
    MU = IntArray(MU_MAX + 1)
    val sqrt = sqrt(MU_MAX.toDouble()).toInt()
    for (i in 0 until MU_MAX) {
        MU!![i] = 1
    }
    for (i in 2..sqrt) {
        if (MU!![i] == 1) {
            //  for each factor found, swap + and -
            for (j in i..MU_MAX step i) {
                MU!![j] *= -i
            }
            //  square factor = 0
            for (j in i * i..MU_MAX step i * i) {
                MU!![j] = 0
            }
        }
    }
    for (i in 2..MU_MAX) {
        when {
            MU!![i] == i -> {
                MU!![i] = 1
            }
            MU!![i] == -i -> {
                MU!![i] = -1
            }
            MU!![i] < 0 -> {
                MU!![i] = 1
            }
            MU!![i] > 0 -> {
                MU!![i] = -1
            }
        }
    }
    return MU!![n]
}
Output:
First 199 terms of the möbius function are as follows:
     1  -1  -1   0  -1   1  -1   0   0   1  -1   0  -1   1   1   0  -1   0  -1  
 0   1   1  -1   0   0   1   0   0  -1  -1  -1   0   1   1   1   0  -1   1   1  
 0  -1  -1  -1   0   0   1  -1   0   0   0   1   0  -1   0   1   0   1   1  -1  
 0  -1   1   0   0   1  -1  -1   0   1  -1  -1   0  -1   1   0   0   1  -1  -1  
 0   0   1  -1   0   1   1   1   0  -1   0   1   0   1   1   1   0  -1   0   0  
 0  -1  -1  -1   0  -1   1  -1   0  -1  -1   1   0  -1  -1   1   0   0   1   1  
 0   0   1   1   0   0   0  -1   0   1  -1  -1   0   1   1   0   0  -1  -1  -1  
 0   1   1   1   0   1   1   0   0  -1   0  -1   0   0  -1   1   0  -1   1   1  
 0   1   0  -1   0  -1   1  -1   0   0  -1   0   0  -1  -1   0   0   1   1  -1  
 0  -1  -1   1   0   1  -1   1   0   0  -1  -1   0  -1   1  -1   0  -1   0  -1  

Lua

Translation of: C
function buildArray(size, value)
    local tbl = {}
    for i=1, size do
        table.insert(tbl, value)
    end
    return tbl
end

MU_MAX = 1000000
sqroot = math.sqrt(MU_MAX)
mu = buildArray(MU_MAX, 1)

for i=2, sqroot do
    if mu[i] == 1 then
        -- for each factor found, swap + and -
        for j=i, MU_MAX, i do
            mu[j] = mu[j] * -i
        end
        -- square factor = 0
        for j=i*i, MU_MAX, i*i do
            mu[j] = 0
        end
    end
end

for i=2, MU_MAX do
    if mu[i] == i then
        mu[i] = 1
    elseif mu[i] == -i then
        mu[i] = -1
    elseif mu[i] < 0 then
        mu[i] = 1
    elseif mu[i] > 0 then
        mu[i] = -1
    end
end

print("First 199 terms of the mobius function are as follows:")
io.write("    ")
for i=1, 199 do
    io.write(string.format("%2d  ", mu[i]))
    if (i + 1) % 20 == 0 then
        print()
    end
end
Output:
First 199 terms of the mobius function are as follows:
     1  -1  -1   0  -1   1  -1   0   0   1  -1   0  -1   1   1   0  -1   0  -1
 0   1   1  -1   0   0   1   0   0  -1  -1  -1   0   1   1   1   0  -1   1   1
 0  -1  -1  -1   0   0   1  -1   0   0   0   1   0  -1   0   1   0   1   1  -1
 0  -1   1   0   0   1  -1  -1   0   1  -1  -1   0  -1   1   0   0   1  -1  -1
 0   0   1  -1   0   1   1   1   0  -1   0   1   0   1   1   1   0  -1   0   0
 0  -1  -1  -1   0  -1   1  -1   0  -1  -1   1   0  -1  -1   1   0   0   1   1
 0   0   1   1   0   0   0  -1   0   1  -1  -1   0   1   1   0   0  -1  -1  -1
 0   1   1   1   0   1   1   0   0  -1   0  -1   0   0  -1   1   0  -1   1   1
 0   1   0  -1   0  -1   1  -1   0   0  -1   0   0  -1  -1   0   0   1   1  -1
 0  -1  -1   1   0   1  -1   1   0   0  -1  -1   0  -1   1  -1   0  -1   0  -1

Mathematica/Wolfram Language

Grid[Partition[MoebiusMu[Range[99]], UpTo[10]]]
Output:
1	-1	-1	0	-1	1	-1	0	0	1
-1	0	-1	1	1	0	-1	0	-1	0
1	1	-1	0	0	1	0	0	-1	-1
-1	0	1	1	1	0	-1	1	1	0
-1	-1	-1	0	0	1	-1	0	0	0
1	0	-1	0	1	0	1	1	-1	0
-1	1	0	0	1	-1	-1	0	1	-1
-1	0	-1	1	0	0	1	-1	-1	0
0	1	-1	0	1	1	1	0	-1	0
1	0	1	1	1	0	-1	0	0	

Nim

Uses the prime decomposition method from https://rosettacode.org/wiki/Prime_decomposition#Nim

import std/[math, sequtils, strformat]

func getStep(n: int): int {.inline.} =
  result = 1 + n shl 2 - n shr 1 shl 1
 
func primeFac(n: int): seq[int] =
  var 
    maxq = int(sqrt(float(n)))
    d = 1
    q: int = 2 + (n and 1)   # Start with 2 or 3 according to oddity.

  while q <= maxq and n %% q != 0:
    q = getStep(d)
    inc d
  if q <= maxq:
    let q1 = primeFac(n /% q)
    let q2 = primeFac(q)
    result = concat(q2, q1, result)
  else:
    result.add(n)

func squareFree(num: int): bool = 
  let fact = primeFac num

  for i in fact:
    if fact.count(i) > 1:
      return false

  return true

func mobius(num: int): int = 
  if num == 1: return num

  let fact = primeFac num

  for i in fact: 
    ## check if it has a squared prime factor
    if fact.count(i) == 2: 
      return 0

  if num.squareFree:
    if fact.len mod 2 == 0: 
      return 1
    else:
      return -1

when isMainModule:
  echo "The first 199 möbius numbers are:" 

  for i in 1..199:
    stdout.write fmt"{mobius(i):4}"
    if i mod 20 == 0:
      echo "" # print newline
Output:
The first 199 möbius numbers are:
   1  -1  -1   0  -1   1  -1   0   0   1  -1   0  -1   1   1   0  -1   0  -1   0
   1   1  -1   0   0   1   0   0  -1  -1  -1   0   1   1   1   0  -1   1   1   0
  -1  -1  -1   0   0   1  -1   0   0   0   1   0  -1   0   1   0   1   1  -1   0
  -1   1   0   0   1  -1  -1   0   1  -1  -1   0  -1   1   0   0   1  -1  -1   0
   0   1  -1   0   1   1   1   0  -1   0   1   0   1   1   1   0  -1   0   0   0
  -1  -1  -1   0  -1   1  -1   0  -1  -1   1   0  -1  -1   1   0   0   1   1   0
   0   1   1   0   0   0  -1   0   1  -1  -1   0   1   1   0   0  -1  -1  -1   0
   1   1   1   0   1   1   0   0  -1   0  -1   0   0  -1   1   0  -1   1   1   0
   1   0  -1   0  -1   1  -1   0   0  -1   0   0  -1  -1   0   0   1   1  -1   0
  -1  -1   1   0   1  -1   1   0   0  -1  -1   0  -1   1  -1   0  -1   0  -1   0

Pascal

 See Mertens_function#Pascal

Perl

use utf8;
use strict;
use warnings;
use feature 'say';
use List::Util 'uniq';

sub prime_factors {
    my ($n, $d, @factors) = (shift, 1);
    while ($n > 1 and $d++) {
        $n /= $d, push @factors, $d until $n % $d;
    }
    @factors
}

sub μ {
    my @p = prime_factors(shift);
    @p == uniq(@p) ? 0 == @p%2 ? 1 : -1 : 0;
}

my @möebius;
push @möebius, μ($_) for 1 .. (my $upto = 199);

say "Möbius sequence - First $upto terms:\n" .
    (' 'x4 . sprintf "@{['%4d' x $upto]}", @möebius) =~ s/((.){80})/$1\n/gr;
Output:
Möbius sequence - First 199 terms:
       1  -1  -1   0  -1   1  -1   0   0   1  -1   0  -1   1   1   0  -1   0  -1
   0   1   1  -1   0   0   1   0   0  -1  -1  -1   0   1   1   1   0  -1   1   1
   0  -1  -1  -1   0   0   1  -1   0   0   0   1   0  -1   0   1   0   1   1  -1
   0  -1   1   0   0   1  -1  -1   0   1  -1  -1   0  -1   1   0   0   1  -1  -1
   0   0   1  -1   0   1   1   1   0  -1   0   1   0   1   1   1   0  -1   0   0
   0  -1  -1  -1   0  -1   1  -1   0  -1  -1   1   0  -1  -1   1   0   0   1   1
   0   0   1   1   0   0   0  -1   0   1  -1  -1   0   1   1   0   0  -1  -1  -1
   0   1   1   1   0   1   1   0   0  -1   0  -1   0   0  -1   1   0  -1   1   1
   0   1   0  -1   0  -1   1  -1   0   0  -1   0   0  -1  -1   0   0   1   1  -1
   0  -1  -1   1   0   1  -1   1   0   0  -1  -1   0  -1   1  -1   0  -1   0  -1

Phix

with javascript_semantics
function Moebius(integer n)
    if n=1 then return 1 end if
    sequence f = prime_factors(n,true)
    for i=2 to length(f) do
        if f[i] = f[i-1] then return 0 end if
    end for
    return iff(odd(length(f))?-1:+1)
end function

sequence s = {"  ."}
for i=1 to 199 do s = append(s,sprintf("%3d",Moebius(i))) end for
puts(1,join_by(s,1,20," "))
Output:
  .   1  -1  -1   0  -1   1  -1   0   0   1  -1   0  -1   1   1   0  -1   0  -1
  0   1   1  -1   0   0   1   0   0  -1  -1  -1   0   1   1   1   0  -1   1   1
  0  -1  -1  -1   0   0   1  -1   0   0   0   1   0  -1   0   1   0   1   1  -1
  0  -1   1   0   0   1  -1  -1   0   1  -1  -1   0  -1   1   0   0   1  -1  -1
  0   0   1  -1   0   1   1   1   0  -1   0   1   0   1   1   1   0  -1   0   0
  0  -1  -1  -1   0  -1   1  -1   0  -1  -1   1   0  -1  -1   1   0   0   1   1
  0   0   1   1   0   0   0  -1   0   1  -1  -1   0   1   1   0   0  -1  -1  -1
  0   1   1   1   0   1   1   0   0  -1   0  -1   0   0  -1   1   0  -1   1   1
  0   1   0  -1   0  -1   1  -1   0   0  -1   0   0  -1  -1   0   0   1   1  -1
  0  -1  -1   1   0   1  -1   1   0   0  -1  -1   0  -1   1  -1   0  -1   0  -1

Python

Everything verbatim from: https://www.geeksforgeeks.org/program-mobius-function/

All code by: Manish Shaw

Method 1

We iterate through all numbers i smaller than or equal to N. For every number we check if it divides N. If yes, we check if it’s also prime. If both conditions are satisfied, we check if its square also divides N. If yes, we return 0. If the square doesn’t divide, we increment count of prime factors. Finally, we return 1 if there are an even number of prime factors and return -1 if there are odd number of prime factors.


# Python Program to evaluate
# Mobius def M(N) = 1 if N = 1
# M(N) = 0 if any prime factor
# of N is contained twice
# M(N) = (-1)^(no of distinct
# prime factors)
# Python Program to
# evaluate Mobius def
# M(N) = 1 if N = 1
# M(N) = 0 if any
# prime factor of
# N is contained twice
# M(N) = (-1)^(no of
# distinct prime factors)
 
# def to check if
# n is prime or not
def isPrime(n) :
 
    if (n < 2) :
        return False
    for i in range(2, n + 1) :
        if (i * i <= n and n % i == 0) :
            return False
    return True
 
def mobius(N) :
     
    # Base Case
    if (N == 1) :
        return 1
 
    # For a prime factor i
    # check if i^2 is also
    # a factor.
    p = 0
    for i in range(1, N + 1) :
        if (N % i == 0 and
                isPrime(i)) :
 
            # Check if N is
            # divisible by i^2
            if (N % (i * i) == 0) :
                return 0
            else :
 
                # i occurs only once,
                # increase f
                p = p + 1
 
    # All prime factors are
    # contained only once
    # Return 1 if p is even
    # else -1
    if(p % 2 != 0) :
        return -1
    else :
        return 1
 
# Driver Code
print("Mobius numbers from 1..99:")
      
for i in range(1, 100):
  print(f"{mobius(i):>4}", end = '')

  if i % 20 == 0: print()
# This code is contributed by
# Manish Shaw(manishshaw1)
Output:
Mobius numbers from 1..99:
   1  -1  -1   0  -1   1  -1   0   0   1  -1   0  -1   1   1   0  -1   0  -1   0
   1   1  -1   0   0   1   0   0  -1  -1  -1   0   1   1   1   0  -1   1   1   0
  -1  -1  -1   0   0   1  -1   0   0   0   1   0  -1   0   1   0   1   1  -1   0
  -1   1   0   0   1  -1  -1   0   1  -1  -1   0  -1   1   0   0   1  -1  -1   0
   0   1  -1   0   1   1   1   0  -1   0   1   0   1   1   1   0  -1   0   0

Method 2 (Efficient)

The idea is based on efficient program to print all prime factors of a given number. The interesting thing is, we do not need inner while loop here because if a number divides more than once, we can immediately return 0.

# Python Program to evaluate
# Mobius def M(N) = 1 if N = 1
# M(N) = 0 if any prime factor
# of N is contained twice
# M(N) = (-1)^(no of distinct
# prime factors)
import math
 
# def to check if n
# is prime or not
def isPrime(n) :
 
    if (n < 2) :
        return False
    for i in range(2, n + 1) :
        if (n % i == 0) :
            return False
        i = i * i
    return True
 
def mobius(n) :
 
    p = 0
 
    # Handling 2 separately
    if (n % 2 == 0) :
     
        n = int(n / 2)
        p = p + 1
 
        # If 2^2 also
        # divides N
        if (n % 2 == 0) :
            return 0
     
 
    # Check for all
    # other prime factors
    for i in range(3, int(math.sqrt(n)) + 1) :
     
        # If i divides n
        if (n % i == 0) :
         
            n = int(n / i)
            p = p + 1
 
            # If i^2 also
            # divides N
            if (n % i == 0) :
                return 0
        i = i + 2   
     
    if(p % 2 == 0) :
        return -1
    else :
        return 1
 
# Driver Code
print("Mobius numbers from 1..99:")
      
for i in range(1, 100):
  print(f"{mobius(i):>4}", end = '')
# This code is contributed by
# Manish Shaw(manishshaw1)
Output:
Mobius numbers from 1..99:
  -1   1  -1   0  -1   1  -1   0   0   1  -1   0  -1   1   1   0  -1   0  -1   0
   1   1  -1   0   0   1   0   0  -1  -1  -1   0   1   1   1   0  -1   1   1   0
  -1  -1  -1   0   0   1  -1   0   0   0   1   0  -1   0   1   0   1   1  -1   0
  -1   1   0   0   1  -1  -1   0   1  -1  -1   0  -1   1   0   0   1  -1  -1   0
   0   1  -1   0   1   1   1   0  -1   0   1   0   1   1   1   0  -1   0   0

Raku

(formerly Perl 6)

Works with: Rakudo version 2019.11

Möbius number is not defined for n == 0. Raku arrays are indexed from 0 so store a blank value at position zero to keep n and μ(n) aligned.

use Prime::Factor;

sub μ (Int \n) {
    return 0 if n %% 4 or n %% 9 or n %% 25 or n %% 49 or n %% 121;
    my @p = prime-factors(n);
    +@p == +@p.unique ?? +@p %% 2 ?? 1 !! -1 !! 0
}

my @möbius = lazy flat '', 1, (2..*).hyper.map: -> \n { μ(n) };

# The Task
put "Möbius sequence - First 199 terms:\n",
    @möbius[^200]».fmt('%3s').batch(20).join: "\n";
Output:
Möbius sequence - First 199 terms:
      1  -1  -1   0  -1   1  -1   0   0   1  -1   0  -1   1   1   0  -1   0  -1
  0   1   1  -1   0   0   1   0   0  -1  -1  -1   0   1   1   1   0  -1   1   1
  0  -1  -1  -1   0   0   1  -1   0   0   0   1   0  -1   0   1   0   1   1  -1
  0  -1   1   0   0   1  -1  -1   0   1  -1  -1   0  -1   1   0   0   1  -1  -1
  0   0   1  -1   0   1   1   1   0  -1   0   1   0   1   1   1   0  -1   0   0
  0  -1  -1  -1   0  -1   1  -1   0  -1  -1   1   0  -1  -1   1   0   0   1   1
  0   0   1   1   0   0   0  -1   0   1  -1  -1   0   1   1   0   0  -1  -1  -1
  0   1   1   1   0   1   1   0   0  -1   0  -1   0   0  -1   1   0  -1   1   1
  0   1   0  -1   0  -1   1  -1   0   0  -1   0   0  -1  -1   0   0   1   1  -1
  0  -1  -1   1   0   1  -1   1   0   0  -1  -1   0  -1   1  -1   0  -1   0  -1

REXX

Note that the   Möbius   function is also spelled   Mobius   and/or   Moebius,   and it is also known as the   mu   function,   where   mu   is the Greek symbol   μ.

Programming note:   This REXX version supports the specifying of the low and high values to be generated,
as well as the group size for the grid   (it can be specified as   1   which will show a vertical list).

A null value will be shown as a bullet (•) when showing the Möbius value of for zero   (this can be changed in the 2nd line of the   mobius   function).

The above "feature" was added to make the grid to be aligned with other solutions.

The function to computer some prime numbers is a bit of an overkill, but the goal was to keep it general  (in case of larger/higher ranges for a Möbius sequence).

/*REXX pgm computes & shows a value grid of the Möbius function for a range of integers.*/
parse arg LO HI grp .                            /*obtain optional arguments from the CL*/
if  LO=='' |  LO==","  then  LO=   0             /*Not specified?  Then use the default.*/
if  HI=='' |  HI==","  then  HI= 199             /* "      "         "   "   "     "    */
if grp=='' | grp==","  then grp=  20             /* "      "         "   "   "     "    */
                                                 /*                            ______   */
call genP HI                                     /*generate primes up to the  √  HI     */
say center(' The Möbius sequence from ' LO " ──► " HI" ", max(50, grp*3), '═')   /*title*/
$=                                               /*variable holds output grid of GRP #s.*/
    do j=LO  to  HI;  $= $  right( mobius(j), 2) /*process some numbers from  LO ──► HI.*/
    if words($)==grp  then do;  say substr($, 2);  $=    /*show grid if fully populated,*/
                           end                           /*  and nullify it for more #s.*/
    end   /*j*/                                  /*for small grids, using wordCnt is OK.*/

if $\==''  then say substr($, 2)                 /*handle any residual numbers not shown*/
exit                                             /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
mobius: procedure expose @.;  parse arg x        /*obtain a integer to be tested for mu.*/
        if x<1  then return '∙'                  /*special? Then return symbol for null.*/
        #= 0                                     /*start with a value of zero.          */
             do k=1;  p= @.k                     /*get the  Kth  (pre─generated)  prime.*/
             if p>x  then leave                  /*prime (P)   > X?    Then we're done. */
             if p*p>x  then do;   #= #+1;  leave /*prime (P**2 > X?    Bump # and leave.*/
                            end
             if x//p==0  then do; #= #+1         /*X divisible by P?   Bump mu number.  */
                                  x= x % p       /*                    Divide by prime. */
                                  if x//p==0  then return 0  /*X÷by P?  Then return zero*/
                              end
             end   /*k*/                         /*#  (below) is almost always small, <9*/
        if #//2==0  then return  1               /*Is # even?   Then return postive  1  */
                         return -1               /* " "  odd?     "     "   negative 1. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
genP: @.1=2; @.2=3; @.3=5; @.4=7; @.5=11; @.6= 13;  nP=6  /*assign low primes; # primes.*/
                    do lim=nP  until lim*lim>=HI /*only keep primes up to the  sqrt(HI).*/
                    end   /*lim*/
       do j=@.nP+4  by 2  to HI                  /*only find odd primes from here on.   */
       parse var j '' -1 _;if _==5  then iterate /*Is last digit a "5"?   Then not prime*/
       if j// 3==0  then iterate                 /*is J divisible by  3?    "   "    "  */
       if j// 7==0  then iterate                 /* " "     "      "  7?    "   "    "  */
       if j//11==0  then iterate                 /* " "     "      " 11?    "   "    "  */
       if j//13==0  then iterate                 /* " "     "      " 13?    "   "    "  */
                 do k=7  while k*k<=j            /*divide by some generated odd primes. */
                 if j // @.k==0  then iterate j  /*Is J divisible by  P?  Then not prime*/
                 end   /*k*/                     /* [↓]  a prime  (J)  has been found.  */
       nP= nP+1;    if nP<=HI  then @.nP= j      /*bump prime count; assign prime to  @.*/
       end      /*j*/;              return
output   when using the default inputs:

Output note:   note the use of a bullet (•) to signify that a "null" is being shown (for the 0th entry).

══════════ The Möbius sequence from  0  ──►  199 ═══════════
 ∙  1 -1 -1  0 -1  1 -1  0  0  1 -1  0 -1  1  1  0 -1  0 -1
 0  1  1 -1  0  0  1  0  0 -1 -1 -1  0  1  1  1  0 -1  1  1
 0 -1 -1 -1  0  0  1 -1  0  0  0  1  0 -1  0  1  0  1  1 -1
 0 -1  1  0  0  1 -1 -1  0  1 -1 -1  0 -1  1  0  0  1 -1 -1
 0  0  1 -1  0  1  1  1  0 -1  0  1  0  1  1  1  0 -1  0  0
 0 -1 -1 -1  0 -1  1 -1  0 -1 -1  1  0 -1 -1  1  0  0  1  1
 0  0  1  1  0  0  0 -1  0  1 -1 -1  0  1  1  0  0 -1 -1 -1
 0  1  1  1  0  1  1  0  0 -1  0 -1  0  0 -1  1  0 -1  1  1
 0  1  0 -1  0 -1  1 -1  0  0 -1  0  0 -1 -1  0  0  1  1 -1
 0 -1 -1  1  0  1 -1  1  0  0 -1 -1  0 -1  1 -1  0 -1  0 -1

Ring

Translation of: FreeBASIC
mobStr = "      . "

for i = 1 to 200
    if mobius(i) >= 0
       mobStr + = " "
    ok
    temp = string(mobius(i))
    if left(temp,2) = "-0"
       temp = right(temp,len(temp)-1)
    ok
    mobStr += temp + " "
    if i % 10 = 9  
        see mobStr + nl
        mobStr = "     "
    ok
next

func mobius(n)
     if n = 1 
        return 1
     ok
     for d = 2 to ceil(sqrt(n))
         if n % d = 0  
            if n % (d*d) = 0
               return 0
            ok
            return -mobius(n/d)
         ok
     next 
     return -1

Output:

      .  1 -1 -1  0 -1  1 -1  0  0 
      1 -1  0 -1  1  1  0 -1  0 -1 
      0  1  1 -1  0  0  1  0  0 -1 
     -1 -1  0  1  1  1  0 -1  1  1 
      0 -1 -1 -1  0  0  1 -1  0  0 
      0  1  0 -1  0  1  0  1  1 -1 
      0 -1  1  0  0  1 -1 -1  0  1 
     -1 -1  0 -1  1  0  0  1 -1 -1 
      0  0  1 -1  0  1  1  1  0 -1 
      0  1  0  1  1  1  0 -1  0  0 
      0 -1 -1 -1  0 -1  1 -1  0 -1 
     -1  1  0 -1 -1  1  0  0  1  1 
      0  0  1  1  0  0  0 -1  0  1 
     -1 -1  0  1  1  0  0 -1 -1 -1 
      0  1  1  1  0  1  1  0  0 -1 
      0 -1  0  0 -1  1  0 -1  1  1 
      0  1  0 -1  0 -1  1 -1  0  0 
     -1  0  0 -1 -1  0  0  1  1 -1 
      0 -1 -1  1  0  1 -1  1  0  0 
     -1 -1  0 -1  1 -1  0 -1  0 -1 

Ruby

require 'prime'

def μ(n)
  pd = n.prime_division
  return 0 unless pd.map(&:last).all?(1)
  pd.size.even? ? 1 : -1
end

(["  "] + (1..199).map{|n|"%2s" % μ(n)}).each_slice(20){|line| puts line.join(" ") }
Output:
    1 -1 -1  0 -1  1 -1  0  0  1 -1  0 -1  1  1  0 -1  0 -1
 0  1  1 -1  0  0  1  0  0 -1 -1 -1  0  1  1  1  0 -1  1  1
 0 -1 -1 -1  0  0  1 -1  0  0  0  1  0 -1  0  1  0  1  1 -1
 0 -1  1  0  0  1 -1 -1  0  1 -1 -1  0 -1  1  0  0  1 -1 -1
 0  0  1 -1  0  1  1  1  0 -1  0  1  0  1  1  1  0 -1  0  0
 0 -1 -1 -1  0 -1  1 -1  0 -1 -1  1  0 -1 -1  1  0  0  1  1
 0  0  1  1  0  0  0 -1  0  1 -1 -1  0  1  1  0  0 -1 -1 -1
 0  1  1  1  0  1  1  0  0 -1  0 -1  0  0 -1  1  0 -1  1  1
 0  1  0 -1  0 -1  1 -1  0  0 -1  0  0 -1 -1  0  0  1  1 -1
 0 -1 -1  1  0  1 -1  1  0  0 -1 -1  0 -1  1 -1  0 -1  0 -1

Sidef

Built-in:

say moebius(53)    #=> -1
say moebius(54)    #=> 0
say moebius(55)    #=> 1

Simple implementation:

func μ(n) {
    var f = n.factor_exp.map { .tail }
    f.any { _ > 1 } ? 0 : ((-1)**f.sum)
}

with (199) { |n|
    say "Values of the Möbius function for numbers in the range 1..#{n}:"
    [' '] + (1..n->map(μ)) -> each_slice(20, {|*line|
        say line.map { '%2s' % _ }.join(' ')
    })
}
Output:
Values of the Möbius function for numbers in the range 1..199:
    1 -1 -1  0 -1  1 -1  0  0  1 -1  0 -1  1  1  0 -1  0 -1
 0  1  1 -1  0  0  1  0  0 -1 -1 -1  0  1  1  1  0 -1  1  1
 0 -1 -1 -1  0  0  1 -1  0  0  0  1  0 -1  0  1  0  1  1 -1
 0 -1  1  0  0  1 -1 -1  0  1 -1 -1  0 -1  1  0  0  1 -1 -1
 0  0  1 -1  0  1  1  1  0 -1  0  1  0  1  1  1  0 -1  0  0
 0 -1 -1 -1  0 -1  1 -1  0 -1 -1  1  0 -1 -1  1  0  0  1  1
 0  0  1  1  0  0  0 -1  0  1 -1 -1  0  1  1  0  0 -1 -1 -1
 0  1  1  1  0  1  1  0  0 -1  0 -1  0  0 -1  1  0 -1  1  1
 0  1  0 -1  0 -1  1 -1  0  0 -1  0  0 -1 -1  0  0  1  1 -1
 0 -1 -1  1  0  1 -1  1  0  0 -1 -1  0 -1  1 -1  0 -1  0 -1

Tiny BASIC

Tiny BASIC is not suited for printing tables, so this is limited to prompting for a single number and calculating its Mobius number.

    PRINT "Enter an integer"
    INPUT N
    IF N < 0 THEN LET N = -N
    IF N < 2 THEN GOTO 100 + N
    LET C = 1
    LET F = 2
 10 IF ((N/F)/F)*F*F = N THEN GOTO 100
    IF (N/F)*F = N THEN GOTO 30
 20 LET F = F + 1
    IF F<=N THEN GOTO 10
    GOTO 100 + C
 30 LET N = N / F
    LET C = -C
    GOTO 20
 99 PRINT "-1"
    END
100 PRINT "0"
    END
101 PRINT "1"
    END

Wren

Library: Wren-fmt
Library: Wren-math
import "/fmt" for Fmt
import "/math" for Int

var isSquareFree = Fn.new { |n|
    var i = 2
    while (i * i <= n) {
        if (n%(i*i) == 0) return false
        i = (i > 2) ? i + 2 : i + 1
    }
    return true
}

var mu = Fn.new { |n|
    if (n < 1) Fiber.abort("Argument must be a positive integer")
    if (n == 1) return 1
    var sqFree = isSquareFree.call(n)
    var factors = Int.primeFactors(n)
    if (sqFree && factors.count % 2 == 0) return 1
    if (sqFree) return -1
    return 0
}

System.print("The first 199 Möbius numbers are:")
for (i in 0..9) {
    for (j in 0..19) {
        if (i == 0 && j == 0) {
            System.write("    ")
        } else {
            System.write("%(Fmt.dm(3, mu.call(i*20 + j))) ")
        }
    }
    System.print()
}
Output:
The first 199 Möbius numbers are:
      1  -1  -1   0  -1   1  -1   0   0   1  -1   0  -1   1   1   0  -1   0  -1 
  0   1   1  -1   0   0   1   0   0  -1  -1  -1   0   1   1   1   0  -1   1   1 
  0  -1  -1  -1   0   0   1  -1   0   0   0   1   0  -1   0   1   0   1   1  -1 
  0  -1   1   0   0   1  -1  -1   0   1  -1  -1   0  -1   1   0   0   1  -1  -1 
  0   0   1  -1   0   1   1   1   0  -1   0   1   0   1   1   1   0  -1   0   0 
  0  -1  -1  -1   0  -1   1  -1   0  -1  -1   1   0  -1  -1   1   0   0   1   1 
  0   0   1   1   0   0   0  -1   0   1  -1  -1   0   1   1   0   0  -1  -1  -1 
  0   1   1   1   0   1   1   0   0  -1   0  -1   0   0  -1   1   0  -1   1   1 
  0   1   0  -1   0  -1   1  -1   0   0  -1   0   0  -1  -1   0   0   1   1  -1 
  0  -1  -1   1   0   1  -1   1   0   0  -1  -1   0  -1   1  -1   0  -1   0  -1 

XPL0

func Mobius(N);
int  N, Cnt, F, K;
[Cnt:= 0;
F:= 2;  K:= 0;
repeat  if rem(N/F) = 0 then
                [Cnt:= Cnt+1;
                N:= N/F;
                K:= K+1;
                if K >= 2 then return 0;
                ]
        else    [F:= F+1;  K:= 0];
until   F > N;
return if Cnt&1 then -1 else 1;
];

int  N;
[Format(3, 0);
Text(0, "   ");
for N:= 1 to 199 do
        [RlOut(0, float(Mobius(N)));
        if rem(N/20) = 19 then CrLf(0);
        ];
]
Output:
     1 -1 -1  0 -1  1 -1  0  0  1 -1  0 -1  1  1  0 -1  0 -1
  0  1  1 -1  0  0  1  0  0 -1 -1 -1  0  1  1  1  0 -1  1  1
  0 -1 -1 -1  0  0  1 -1  0  0  0  1  0 -1  0  1  0  1  1 -1
  0 -1  1  0  0  1 -1 -1  0  1 -1 -1  0 -1  1  0  0  1 -1 -1
  0  0  1 -1  0  1  1  1  0 -1  0  1  0  1  1  1  0 -1  0  0
  0 -1 -1 -1  0 -1  1 -1  0 -1 -1  1  0 -1 -1  1  0  0  1  1
  0  0  1  1  0  0  0 -1  0  1 -1 -1  0  1  1  0  0 -1 -1 -1
  0  1  1  1  0  1  1  0  0 -1  0 -1  0  0 -1  1  0 -1  1  1
  0  1  0 -1  0 -1  1 -1  0  0 -1  0  0 -1 -1  0  0  1  1 -1
  0 -1 -1  1  0  1 -1  1  0  0 -1 -1  0 -1  1 -1  0 -1  0 -1

Yabasic

Translation of: FreeBASIC
outstr$ = " .  "
for i = 1 to 200
    if mobius(i) >= 0 then outstr$ = outstr$ + " " : fi
    outstr$ = outstr$ + str$(mobius(i)) + "  "
    if mod(i, 10) = 9 then 
        print outstr$
        outstr$ = ""
    end if
next i
end

sub mobius(n)
    if n = 1 then return 1 : fi
    for d = 2 to int(sqr(n))
        if mod(n, d) = 0 then 
            if mod(n, (d*d)) = 0 then return 0 : fi
            return -mobius(n/d)
        end if
    next d
    return -1
end sub

zkl

fcn mobius(n){
   pf:=primeFactors(n);
   sq:=pf.filter1('wrap(f){ (n % (f*f))==0 });  // False if square free
   if(sq==False){ if(pf.len().isEven) 1 else -1 }
   else 0
}
fcn primeFactors(n){  // Return a list of prime factors of n
   acc:=fcn(n,k,acc,maxD){  // k is 2,3,5,7,9,... not optimum
      if(n==1 or k>maxD) acc.close();
      else{
	 q,r:=n.divr(k);   // divr-->(quotient,remainder)
	 if(r==0) return(self.fcn(q,k,acc.write(k),q.toFloat().sqrt()));
	 return(self.fcn(n,k+1+k.isOdd,acc,maxD))  # both are tail recursion
      }
   }(n,2,Sink(List),n.toFloat().sqrt());
   m:=acc.reduce('*,1);      // mulitply factors
   if(n!=m) acc.append(n/m); // opps, missed last factor
   else acc;
}
[1..199].apply(mobius)
.pump(Console.println, T(Void.Read,19,False),
	fcn{ vm.arglist.pump(String,"%3d".fmt) });
Output:
  1 -1 -1  0 -1  1 -1  0  0  1 -1  0 -1  1  1  0 -1  0 -1  0
  1  1 -1  0  0  1  0  0 -1 -1 -1  0  1  1  1  0 -1  1  1  0
 -1 -1 -1  0  0  1 -1  0  0  0  1  0 -1  0  1  0  1  1 -1  0
 -1  1  0  0  1 -1 -1  0  1 -1 -1  0 -1  1  0  0  1 -1 -1  0
  0  1 -1  0  1  1  1  0 -1  0  1  0  1  1  1  0 -1  0  0  0
 -1 -1 -1  0 -1  1 -1  0 -1 -1  1  0 -1 -1  1  0  0  1  1  0
  0  1  1  0  0  0 -1  0  1 -1 -1  0  1  1  0  0 -1 -1 -1  0
  1  1  1  0  1  1  0  0 -1  0 -1  0  0 -1  1  0 -1  1  1  0
  1  0 -1  0 -1  1 -1  0  0 -1  0  0 -1 -1  0  0  1  1 -1  0
 -1 -1  1  0  1 -1  1  0  0 -1 -1  0 -1  1 -1  0 -1  0 -1