Möbius function
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
Factor
The mobius word exists in the math.extras vocabulary. See the implementation here.
<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
Julia
<lang 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
- =
</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
Perl 6
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) };
- 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
REXX
<lang rexx>/*REXX pgm computes and shows a value grid of the Möbius funtion 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==0 then return '∙' /*Zero? Then return a symbol for null.*/
if x==1 then return 1 /*Unity? Then return unity. */
#= 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*/
return -1 ** # /*raise negative unity to the mu power.*/
/*──────────────────────────────────────────────────────────────────────────────────────*/ 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; end /*only keep primes up to the sqrt(HI). */
do j=@.nP+4 by 2 to lim /*only find odd primes from here on. */
if j// 3==0 then iterate /*is J divisible by #3 Then not prime*/
parse var j -1 _;if _==5 then iterate /*Is last digit a "5"? " " " */
if j// 7==0 then iterate /*is J divisible by 7? " " " */
if j//11==0 then iterate /* " " " " 11? " " " */
if j//13==0 then iterate /*is " " " 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<=lim then @.nP=j /*bump prime count; assign prime to @.*/
end /*j*/; return</lang>
- 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