Möbius function

Möbius function is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

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) as the sum of the primitive factors. It has the values {−1, 0, 1} depending on the factorization of n:

• 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

• Wikipedia: Möbius function

Related Tasks

• Mertens function

Factor

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

Works with: Factor version 0.99 2020-01-23

<lang factor>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</lang>

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

Go

<lang 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])
   }

}</lang>

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

Perl 6

Works with: Rakudo version 2019.11

Möbius number is not defined for n == 0. Perl 6 arrays are indexed from 0 so store a blank value at position zero to keep n and μ(n) aligned.

<lang perl6>use Prime::Factor;

sub μ (Int \n) {

   my @p = prime-factors(n);
   +@p == +Bag(@p).keys ?? +@p %% 2 ?? 1 !! -1 !! 0

}

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

1. The Task

put "Möbius sequence - First 199 terms:\n",

   @möbius[^200]».fmt('%3s').rotor(20).join("\n");</lang>

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