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Ingy döt Net 2023-07-01 11:58:00 -04:00
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Task/Mertens-function/00-META.yaml Normal file
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---
from: http://rosettacode.org/wiki/Mertens_function
note: Prime Numbers

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The Mertens function '''M(x)''' is the '''count''' of '''square-free integers''' up to '''x''' that have an '''even number''' of prime factors, '''minus''' the '''count''' of those that have an '''odd number'''.
It is an extension of the [[Möbius function]]. Given the Möbius function '''μ(n)''', the Mertens function '''M(x)''' is the sum of the Möbius numbers from '''n == 1''' through '''n == x'''.
;Task:
:* Write a routine (function, procedure, whatever) to find the Mertens number for any positive integer '''x'''.
:* 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.)
:* Use that routine to find and display here, on this page, the number of times the Mertens function sequence '''is equal to zero''' in the range '''M(1)''' through '''M(1000)'''.
:* Use that routine to find and display here, on this page, the number of times the Mertens function sequence '''crosses zero''' in the range '''M(1)''' through '''M(1000)'''. (Crossing defined as this term equal to zero but preceding term '''not'''.)
;See also:
:* [[wp:Mertens function|Wikipedia: Mertens function]]
:* [[wp:Möbius function|Wikipedia: Möbius function]]
:* [[oeis:A002321|OEIS: A002321 - Mertens's function]]
:* [[oeis:A028442|OEIS: A028442 - Numbers n such that Mertens's function M(n) is zero]]
:* [https://www.youtube.com/watch?v=uvMGZb0Suyc Numberphile - Mertens Conjecture]
:* [https://codegolf.stackexchange.com/questions/94814/compute-the-mertens-function Stackexchange: compute the mertens function]
''This is not code golf.   The stackexchange link is provided as an algorithm reference, not as a guide.''
;Related tasks:
:* [[Möbius function]]
<br><br>

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F mertens(count)
‘Generate Mertens numbers’
V m = [-1, 1]
L(n) 2 .. count
m.append(1)
L(k) 2 .. n
m[n] -= m[n I/ k]
R m
V ms = mertens(1000)
print(‘The first 99 Mertens numbers are:’)
print(‘ ’, end' ‘ ’)
V col = 1
L(n) ms[1.<100]
print(‘#2’.format(n), end' ‘ ’)
col++
I col == 10
print()
col = 0
V zeroes = sum(ms.map(x -> Int(x == 0)))
V crosses = sum(zip(ms, ms[1..]).map((a, b) -> Int(a != 0 & b == 0)))
print(‘M(N) equals zero #. times.’.format(zeroes))
print(‘M(N) crosses zero #. times.’.format(crosses))

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* Mertens function - 01/05/2023
MERTENS CSECT
USING MERTENS,R13 base register
B 72(R15) skip savearea
DC 17F'0' savearea
SAVE (14,12) save previous context
ST R13,4(R15) link backward
ST R15,8(R13) link forward
LR R13,R15 set addressability
LA R0,1 1
STH R0,MM m(1)=1
LA R6,2 i=2
DO WHILE=(CH,R6,LE,=AL2(NN)) do i=2 to n
LR R1,R6 i
SLA R1,1 *2 (H)
LA R0,1 1
STH R0,MM-2(R1) m(i)=1
LA R7,2 j=2
DO WHILE=(CR,R7,LE,R6) do j=2 to i
LR R4,R6 i
SRDA R4,32 ~
LR R1,R7 j
DR R4,R1 i/j
LR R8,R5 d=i/j
LR R4,R6 i
SLA R4,1 *2 (H)
LH R2,MM-2(R4) m(i)
LR R1,R8 d
SLA R1,1 *2 (H)
LH R3,MM-2(R1) m(d)
SR R2,R3 m(i)-m(d)
STH R2,MM-2(R4) m(i)=m(i)-m(d)
LA R7,1(R7) j++
ENDDO , enddo j
LA R6,1(R6) i++
ENDDO , enddo i
XPRNT =C'the first 99 Mertens numbers are:',34 print buffer
LA R9,PG @buffer=pg
MVC PG,=CL80' ' clean buffer
MVC 0(3,R9),=CL3' ' output ' '
LA R9,3(R9) @buffer+=3
LA R7,9 j=9
LA R6,1 i=1
DO WHILE=(CH,R6,LE,=AL2(99)) do i=1 to 99
LR R1,R6 i
SLA R1,1 *2 (H)
LH R2,MM-2(R1) m(i)
XDECO R2,XDEC edit m(i)
MVC 0(3,R9),XDEC+9 output m(i)
LA R9,3(R9) @buffer+=3
BCTR R7,0 j=j-1
IF LTR,R7,Z,R7 THEN if j=0 then do;
LA R7,10 j=10
XPRNT PG,L'PG print buffer
LA R9,PG @buffer=pg
ENDIF , endif
LA R6,1(R6) i++
ENDDO , enddo i
SR R10,R10 zero=0
SR R11,R11 cross=0
LA R6,1 i=2
DO WHILE=(CH,R6,LE,=AL2(NN)) do i=2 to n
LR R1,R6 i
SLA R1,1 *2 (H)
LH R2,MM-2(R1) m(i)
IF LTR,R2,Z,R2 THEN if m(i)=0 then
LA R10,1(R10) zero=zero+1
LR R1,R6 i
BCTR R1,0 i-1
SLA R1,1 *2 (H)
LH R2,MM-2(R1) m(i-1)
IF LTR,R2,NZ,R2 THEN if m(i-1)^=0 then
LA R11,1(R11) cross=cross+1
ENDIF , endif
ENDIF , endif
LA R6,1(R6) i++
ENDDO , enddo i
MVC PG,=CL80' ' clean buffer
MVC PG(13),=C'm(i) is zero '
XDECO R10,XDEC edit zero
MVC PG+13(2),XDEC+10 output zero
MVC PG+15(7),=C' times.'
XPRNT PG,L'PG print buffer
MVC PGI,=H'0'
MVC PG,=CL80' ' clean buffer
MVC PG(18),=C'm(i) crosses zero '
XDECO R11,XDEC edit cross
MVC PG+18(2),XDEC+10 output cross
MVC PG+20(7),=C' times.'
XPRNT PG,L'PG print buffer
L R13,4(0,R13) restore previous savearea pointer
RETURN (14,12),RC=0 restore registers from calling save
NN EQU 1000 n
PG DS CL80 buffer
PGI DC H'0' buffer index
XDEC DS CL12 temp for xdeci xdeco
MM DS (NN)H m
REGEQU
END MERTENS

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MAX: equ 1000 ; Amount of numbers to generate
org 100h
;;; Generate Mertens numbers
lxi b,1 ; Start at place 1; BC = current Mertens number
lxi h,MM ; First one is 1
dad b
mvi m,1
outer: inx b ; Next Mertens number
lxi h,MM
dad b
mvi m,1 ; Initialize at 1
lxi d,2 ; DE = inner loop counter ('k'), starts at 2
;;; Now we need to find BC/DE, but there is no hardware divide
;;; We also need to be somewhat clever so it doesn't take forever
inner: push d ; Keep both loop counters safe on the stack
push b
xchg ; Divisor in HL
mov d,b ; Dividend in DE
mov e,c
lxi b,100h ; B = counter, C = zero
double: dad h ; Double divisor
inr b ; Increment counter
call cdehl ; Dividend <= divisor?
jnc double ; If so, keep doubling
mov a,b ; Keep counter
mov b,c ; BC = 0
push b ; Push result variable on stakc (initial 0)
mov b,a ; Restore counter
xchg ; HL = dividend, DE = doubled divisor
subtr: mov a,l ; Try HL -= DE
sub e
mov l,a
mov a,h
sbb d
mov h,a
xthl ; Get result accumulator from stack
cmc ; Flip borrow
mov a,l ; Rotate into result
ral
mov l,a
mov a,h
ral
mov h,a
mov a,l ; Retrieve flag
rar
xthl ; Retrieve rest of divisor
jc $+4 ; If borrow,
dad d ; Add dividend back into divisor
xra a ; DE >> 1
ora d
rar
mov d,a
mov a,e
rar
mov e,a
dcr b ; Are we there yet?
jnz subtr ; If not, try another subtraction
pop h ; HL = quotient
;;; Division is done, do lookup and subraction
lxi d,MM ; Look up M[outer/inner]
dad d
mov e,m ; E = M[BC/DE]
pop b ; Restore BC (n)
lxi h,MM
dad b
mov a,m ; A = M[BC]
sub e ; A = M[BC] - M[BC/DE]
mov m,a ; M[BC] = A
pop d ; Restore DE (k)
;;; Update loops
inx d ; k++
call cbcde ; DE <= BC?
jnc inner
lxi h,MAX
call chlbc ; BC <= MAX?
jnc outer
;;; Print table
lxi d,frst99
call puts
lxi h,MM+1 ; Start of Merten numbers
mvi c,9 ; Column counter
table: mov a,m ; Get Merten number
ana a ; Set flags
mvi b,' ' ; Space
jp prtab ; If positive, print space-number-space
mvi b,'-' ; Otherwise, print minus sign
cma ; And negate the number (make positive)
inr a
prtab: adi '0' ; Make ASCII digit
mov d,a ; Keep number
mov a,b ; Print space or minus sign
call putc
mov a,d ; Restore number
call putc ; Print number
mvi a,' ' ; Print space
call putc
dcr c ; Decrement column counter
jnz tnext
lxi d,nl ; End of columns - print newline
call puts
mvi c,10 ; Column counter
tnext: inx h ; Table done?
mov a,l
cpi 100
jnz table ; If not, keep going
;;; Find zeroes and crossings
lxi b,0 ; B=zeroes, C=crossings
lxi d,MAX ; Counter
lxi h,MM+1
count: mov a,m ; Get number
ana a ; Zero?
jnz cnext
inr b ; If so, add zero
dcx h ; Previous number also zero?
mov a,m
inx h
ana a
jz cnext
inr c ; If not, add crossiong
cnext: inx h
dcx d
mov a,d
ora e
jnz count
lxi d,zero ; Print zeroes
call puts
mov a,b
call puta
lxi d,cross ; Print crossings
call puts
mov a,c
call puta
lxi d,tms
jmp puts
;;; Print character in A using CP/M, keeping registers
putc: push b
push d
push h
mov e,a
mvi c,2
call 5
jmp resrgs
;;; Print number in A, keeping registers
puta: push b
push d
push h
lxi h,num
putad: mvi c,-1
putal: inr c
sui 10
jnc putal
adi 10+'0'
dcx h
mov m,a
mov a,c
ana a
jnz putad
xchg
mvi c,9
call 5
jmp resrgs
;;; Print string in DE using CP/M, keeping registers
puts: push b
push d
push h
mvi c,9
call 5
resrgs: pop h
pop d
pop b
ret
cdehl: mov a,d
cmp h
rnz
mov a,e
cmp l
ret
cbcde: mov a,b
cmp d
rnz
mov a,c
cmp e
ret
chlbc: mov a,h
cmp b
rnz
mov a,l
cmp c
ret
;;; Strings
db '***'
num: db '$'
frst99: db 'First 99 Mertens numbers:',13,10,' $'
nl: db 13,10,'$'
zero: db 'M(N) is zero $'
cross: db ' times.',13,10,'M(N) crosses zero $'
tms: db ' times.$'
;;; Numbers are stored page-aligned after program
MM: equ ($/256)*256+256

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MAX: equ 1000 ; Amount of Mertens numbers to generate
puts: equ 9 ; MS-DOS syscall to print a string
putch: equ 2 ; MS-DOS syscall to print a character
cpu 8086
org 100h
section .text
;;; Generate Mertens numbers
mov bx,M ; BX = pointer to start of Mertens numbers
mov si,1 ; Current Mertens number
mov [si+bx],byte 1 ; First Mertens number is 1
outer: inc si ; Next Mertens number
mov [si+bx],byte 1 ; Starts out at 1...
mov cx,2 ; CX = from 2 to current number,
inner: mov ax,si ; Divide current number,
xor dx,dx
div cx ; By CX
mov di,ax
mov al,[di+bx] ; Get value at that location
sub [si+bx],al ; Subtract from current number
inc cx
cmp cx,si
jbe inner
cmp si,MAX
jbe outer
;;; Print the table
mov dx,frst99 ; First string
call outstr
mov si,1 ; Start at index 1
mov dh,9 ; Column count
table: mov cl,[si+bx] ; Get item
test cl,cl
mov dl,' '
jns .print ; Positive?
mov dl,'-' ; Otherwise, it is negative,
neg cl ; print ' ' and negate
.print: call putc ; Print space or minus
add cl,'0' ; Add ASCII 0
mov dl,cl
call putc ; Print number
mov dl,' '
call putc ; Print space
dec dh ; One less column left
jnz .next
mov dx,nl ; Print newline
call outstr
mov dh,10
.next: inc si ; Done yet?
cmp si,100
jb table ; If not, print next item from table
;;; Calculate zeroes and crossings
xor cx,cx ; CL = zeroes, CH = crossings
mov si,1
mov al,[si+bx] ; AL = current item
zc: inc si
mov ah,al ; AH = previous item
mov al,[si+bx]
test al,al ; Zero?
jnz .next
inc cx ; Then increment zero counter
test ah,ah ; Previous one also zero?
jz .next
inc ch ; Then increment crossing counter
.next: cmp si,MAX ; Done yet?
jbe zc
;;; Print zeroes and crossings
mov dx,zero
call outstr
mov al,cl
call putal
mov dx,cross
call outstr
mov al,ch
call putal
mov dx,tms
jmp outstr
putc: mov ah,putch ; Print character
int 21h
ret
;;; Print AL in decimal format
putal: mov di,num
.loop: aam ; Extract digit
add al,'0' ; Store digit
dec di
mov [di],al
mov al,ah ; Rest of number
test al,al ; Done?
jnz .loop ; If not, get more digits
mov dx,di ; Otherwise, print string
outstr: mov ah,puts
int 21h
ret
section .data
db '***' ; Number output placeholder
num: db '$'
frst99: db 'First 99 Mertens numbers:',13,10,' $'
nl: db 13,10,'$'
zero: db 'M(N) is zero $'
cross: db ' times.',13,10,'M(N) crosses zero $'
tms: db ' times.$'
section .bss
mm: resb MAX ; Mertens numbers
M: equ mm-1 ; 1-based indexing

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BEGIN # compute values of the Mertens function #
# Generate Mertens numbers #
[ 1 : 1000 ]INT m;
m[ 1 ] := 1;
FOR n FROM 2 TO UPB m DO
m[ n ] := 1;
FOR k FROM 2 TO n DO m[ n ] -:= m[ n OVER k ] OD
OD;
# Print table #
print( ( "The first 99 Mertens numbers are:", newline ) );
print( ( " " ) );
INT k := 9;
FOR n TO 99 DO
print( ( whole( m[ n ], -3 ) ) );
IF ( k -:= 1 ) = 0 THEN
k := 10;
print( ( newline ) )
FI
OD;
# Calculate zeroes and crossings #
INT zero := 0;
INT cross := 0;
FOR n FROM 2 TO UPB m DO
IF m[ n ] = 0 THEN
zero +:= 1;
IF m[ n - 1 ] /= 0 THEN cross +:= 1 FI
FI
OD;
print( ( newline ) );
print( ( "M(N) is zero ", whole( zero, -4 ), " times.", newline ) );
print( ( "M(N) crosses zero ", whole( cross, -4 ), " times.", newline ) )
END

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begin % compute values of the Mertens function %
integer array M ( 1 :: 1000 );
integer k, zero, cross;
% Generate Mertens numbers %
M( 1 ) := 1;
for n := 2 until 1000 do begin
M( n ) := 1;
for k := 2 until n do M( n ) := M( n ) - M( n div k )
end for_n ;
% Print table %
write( "The first 99 Mertens numbers are:" );
write( " " );
k := 9;
for n := 1 until 99 do begin
writeon( i_w := 3, s_w := 0, M( n ) );
k := k - 1;
if k = 0 then begin
k := 10;
write()
end if_k_eq_0
end for_n ;
% Calculate zeroes and crossings %
zero := 0;
cross := 0;
for n :=2 until 1000 do begin
if M( n ) = 0 then begin
zero := zero + 1;
if M( n - 1 ) not = 0 then cross := cross + 1
end if_M_n_eq_0
end for_n ;
write( i_w := 2, s_w := 0, "M(N) is zero ", zero, " times." );
write( i_w := 2, s_w := 0, "M(N) crosses zero ", cross, " times." )
end.

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mertens←{
step ← {⍵,-⍨/⌽1,⍵[⌊n÷1↓⍳n←1+≢⍵]}
m1000 ← step⍣999⊢,1
zero ← m1000+.=0
cross ← +/(~∧1⌽⊢)m1000≠0
⎕←'First 99 Mertens numbers:'
⎕←10 10⍴'∘',m1000
⎕←'M(N) is zero ',(⍕zero),' times.'
⎕←'M(N) crosses zero ',(⍕cross),' times.'
}

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INCLUDE "D2:PRINTF.ACT" ;from the Action! Tool Kit
PROC MertensNumbers(INT ARRAY m INT count)
INT n,k
m(1)=1
FOR n=2 TO count
DO
m(n)=1
FOR k=2 TO n
DO
m(n)==-m(n/k)
OD
OD
RETURN
PROC PrintMertens(INT ARRAY m INT count)
CHAR ARRAY s(6)
INT i,col
PrintF("First %I Mertens numbers:%E ",count)
col=1
FOR i=1 TO count
DO
StrI(m(i),s)
PrintF("%3S",s)
col==+1
IF col=10 THEN
col=0 PutE()
FI
OD
RETURN
PROC Main()
DEFINE MAX="1001"
INT ARRAY m(MAX)
INT i,zeroCnt=[0],crossCnt=[0],prev=[0]
Put(125) PutE() ;clear the screen
PrintF("Calculation of Mertens numbers,%E please wait...")
MertensNumbers(m,MAX)
Put(125) PutE() ;clear the screen
PrintMertens(m,99)
FOR i=1 TO MAX
DO
IF m(i)=0 THEN
zeroCnt==+1
IF prev THEN
crossCnt==+1
FI
FI
prev=m(i)
OD
PrintF("%EM(n) is zero %I times for 1<=n<=%I.%E",zeroCnt,MAX-1)
PrintF("%EM(n) crosses zero %I times for 1<=n<=%I.%E",crossCnt,MAX-1)
RETURN

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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
]
mertens: function [z][sum map 1..z => mobius]
print "The first 99 Mertens numbers are:"
loop split.every:20 [""]++map 1..99 => mertens 'a [
print map a 'item -> pad to :string item 2
]
print ""
mertens1000: map 1..1000 => mertens
print ["Times M(x) is zero between 1 and 1000:" size select mertens1000 => zero?]
crossed: new 0
fold mertens1000 [a,b][if and? zero? b not? zero? a -> inc 'crossed, b]
print ["Times M(x) crosses zero between 1 and 1000:" crossed]

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Task/Mertens-function/AutoHotkey/mertens-function.ahk Normal file
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result := "first 100 terms:`n"
loop 100
result .= SubStr(" " Mertens(A_Index), -1) . (Mod(A_Index, 10) ? " " : "`n")
eqZero := crZero := 0, preced:=1
loop 1000
{
if !(x := Mertens(A_Index))
eqZero++, crZero += preced<>0 ? 1 : 0
preced := x
}
result .= "`nfirst 1000 terms:"
MsgBox, 262144, , % result .= "`nequal to zero : " eqZero "`ncrosses zero : " crZero
return
Mertens(n){
loop % n
result += Möbius(A_Index)
return result
}
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
}

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Task/Mertens-function/BASIC/mertens-function.basic Normal file
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10 DEFINT C,Z,N,K,M: DIM M(1000)
20 M(1)=1
30 FOR N=2 TO 1000
40 M(N)=1
50 FOR K=2 TO N: M(N) = M(N)-M(INT(N/K)): NEXT
60 NEXT
70 PRINT "First 99 Mertens numbers:"
80 PRINT " ";
90 FOR N=1 TO 99
100 PRINT USING "###";M(N);
110 IF N MOD 10 = 9 THEN PRINT
120 NEXT
130 C=0: Z=0
140 FOR N=1 TO 1000
150 IF M(N)=0 THEN Z=Z+1: IF M(N-1)<>0 THEN C=C+1
160 NEXT
170 PRINT "M(N) is zero";Z;"times."
180 PRINT "M(N) crosses zero";C;"times."
190 END

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arraybase 1
dim M(1000)
M[1] = 1
for n = 2 to 1000
M[n] = 1
for k = 2 to n
M[n] = M[n] - M[int(n/k)]
next k
next n
print "First 99 Mertens numbers:"
print " ";
for n = 1 to 99
print rjust(string(M[n]),3);
if n mod 10 = 9 then print
next n
numCruza = 0
numEsCero = 0
for n = 1 to 1000
if M[n] = 0 then
numEsCero += 1
if M[n-1] <> 0 then numCruza += 1
end if
next n
print
print "M(n) is zero "; numEsCero; " times."
print "M(n) crosses zero "; numCruza; " times."

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Task/Mertens-function/BCPL/mertens-function.bcpl Normal file
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get "libhdr"
manifest $( limit = 1000 $)
let mertens(v, max) be
$( v!1 := 1
for n = 2 to max do
$( v!n := 1
for k = 2 to n do
v!n := v!n - v!(n/k)
$)
$)
let start() be
$( let m = vec limit
let eqz, crossz = 0, 0
writes("The first 99 Mertens numbers are:*N")
mertens(m, limit)
for y=0 to 90 by 10 do
$( for x=0 to 9 do
test x+y=0
then writes(" ")
else writed(m!(x+y),3)
wrch('*N')
$)
for x=2 to limit do
if m!x=0 then
$( eqz := eqz + 1
unless m!(x-1)=0 do crossz := crossz + 1
$)
writef("M(N) is zero %N times.*N", eqz)
writef("M(N) crosses zero %N times.*N", crossz)
$)

43
Task/Mertens-function/C++/mertens-function.cpp Normal file
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#include <iomanip>
#include <iostream>
#include <vector>
std::vector<int> mertens_numbers(int max) {
std::vector<int> m(max + 1, 1);
for (int n = 2; n <= max; ++n) {
for (int k = 2; k <= n; ++k)
m[n] -= m[n / k];
}
return m;
}
int main() {
const int max = 1000;
auto m(mertens_numbers(max));
std::cout << "First 199 Mertens numbers:\n";
for (int i = 0, column = 0; i < 200; ++i) {
if (column > 0)
std::cout << ' ';
if (i == 0)
std::cout << " ";
else
std::cout << std::setw(2) << m[i];
++column;
if (column == 20) {
std::cout << '\n';
column = 0;
}
}
int zero = 0, cross = 0, previous = 0;
for (int i = 1; i <= max; ++i) {
if (m[i] == 0) {
++zero;
if (previous != 0)
++cross;
}
previous = m[i];
}
std::cout << "M(n) is zero " << zero << " times for 1 <= n <= 1000.\n";
std::cout << "M(n) crosses zero " << cross << " times for 1 <= n <= 1000.\n";
return 0;
}

53
Task/Mertens-function/C/mertens-function.c Normal file
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#include <stdio.h>
#include <stdlib.h>
int* mertens_numbers(int max) {
int* m = malloc((max + 1) * sizeof(int));
if (m == NULL)
return m;
m[1] = 1;
for (int n = 2; n <= max; ++n) {
m[n] = 1;
for (int k = 2; k <= n; ++k)
m[n] -= m[n/k];
}
return m;
}
int main() {
const int max = 1000;
int* mertens = mertens_numbers(max);
if (mertens == NULL) {
fprintf(stderr, "Out of memory\n");
return 1;
}
printf("First 199 Mertens numbers:\n");
const int count = 200;
for (int i = 0, column = 0; i < count; ++i) {
if (column > 0)
printf(" ");
if (i == 0)
printf(" ");
else
printf("%2d", mertens[i]);
++column;
if (column == 20) {
printf("\n");
column = 0;
}
}
int zero = 0, cross = 0, previous = 0;
for (int i = 1; i <= max; ++i) {
int m = mertens[i];
if (m == 0) {
++zero;
if (previous != 0)
++cross;
}
previous = m;
}
free(mertens);
printf("M(n) is zero %d times for 1 <= n <= %d.\n", zero, max);
printf("M(n) crosses zero %d times for 1 <= n <= %d.\n", cross, max);
return 0;
}

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Task/Mertens-function/CLU/mertens-function.clu Normal file
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% Generate Mertens numbers up to a given limit
mertens = proc (limit: int) returns (array[int])
M: array[int] := array[int]$fill(1,limit,0)
M[1] := 1
for n: int in int$from_to(2,limit) do
M[n] := 1
for k: int in int$from_to(2,n) do
M[n] := M[n] - M[n/k]
end
end
return (M)
end mertens
start_up = proc ()
max = 1000
po: stream := stream$primary_output()
M: array[int] := mertens(max)
stream$putl(po, "The first 99 Mertens numbers are:")
for y: int in int$from_to_by(0,90,10) do
for x: int in int$from_to(0,9) do
stream$putright(po, int$unparse(M[x+y]), 3)
except when bounds:
stream$putright(po, "", 3)
end
end
stream$putl(po, "")
end
eqz: int := 0
crossz: int := 0
for i: int in int$from_to(2,max) do
if M[i]=0 then
eqz := eqz + 1
if M[i-1]~=0 then crossz := crossz + 1 end
end
end
stream$putl(po, "M(N) is zero " || int$unparse(eqz) || " times.")
stream$putl(po, "M(N) crosses zero " || int$unparse(crossz) || " times.")
end start_up

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Task/Mertens-function/COBOL/mertens-function.cobol Normal file
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IDENTIFICATION DIVISION.
PROGRAM-ID. MERTENS.
DATA DIVISION.
WORKING-STORAGE SECTION.
01 VARIABLES.
03 M PIC S99 OCCURS 1000 TIMES.
03 N PIC 9(4).
03 K PIC 9(4).
03 V PIC 9(4).
03 IS-ZERO PIC 99 VALUE 0.
03 CROSS-ZERO PIC 99 VALUE 0.
01 OUTPUT-FORMAT.
03 OUT-ITEM.
05 OUT-NUM PIC -9.
05 FILLER PIC X VALUE SPACE.
03 OUT-LINE PIC X(30) VALUE SPACES.
03 OUT-PTR PIC 99 VALUE 4.
PROCEDURE DIVISION.
BEGIN.
PERFORM GENERATE-MERTENS.
PERFORM WRITE-TABLE.
PERFORM COUNT-ZEROES.
STOP RUN.
GENERATE-MERTENS.
MOVE 1 TO M(1).
PERFORM MERTENS-OUTER-LOOP VARYING N FROM 2 BY 1
UNTIL N IS GREATER THAN 1000.
MERTENS-OUTER-LOOP.
MOVE 1 TO M(N).
PERFORM MERTENS-INNER-LOOP VARYING K FROM 2 BY 1
UNTIL K IS GREATER THAN N.
MERTENS-INNER-LOOP.
DIVIDE N BY K GIVING V.
SUBTRACT M(V) FROM M(N).
WRITE-TABLE.
DISPLAY "The first 99 Mertens numbers are: "
PERFORM WRITE-ITEM VARYING N FROM 1 BY 1
UNTIL N IS GREATER THAN 99.
WRITE-ITEM.
MOVE M(N) TO OUT-NUM.
STRING OUT-ITEM DELIMITED BY SIZE INTO OUT-LINE
WITH POINTER OUT-PTR.
IF OUT-PTR IS EQUAL TO 31,
DISPLAY OUT-LINE,
MOVE 1 TO OUT-PTR.
COUNT-ZEROES.
PERFORM TEST-N-ZERO VARYING N FROM 2 BY 1
UNTIL N IS GREATER THAN 1000.
DISPLAY "M(N) is zero " IS-ZERO " times.".
DISPLAY "M(N) crosses zero " CROSS-ZERO " times.".
TEST-N-ZERO.
IF M(N) IS EQUAL TO ZERO,
ADD 1 TO IS-ZERO,
SUBTRACT 1 FROM N GIVING K,
IF M(K) IS NOT EQUAL TO ZERO,
ADD 1 TO CROSS-ZERO.

63
Task/Mertens-function/Cowgol/mertens-function.cowgol Normal file
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include "cowgol.coh";
const MAX := 1000;
# Table output
sub printtab(n: int8) is
if n<0 then
print_char('-');
n := -n;
else
print_char(' ');
end if;
print_char(n as uint8 + '0');
print_char(' ');
end sub;
# Generate Merten numbers
var M: int8[MAX+1];
M[0] := 0;
M[1] := 1;
var n: @indexof M := 2;
while n < @sizeof M loop
M[n] := 1;
var k: @indexof M := 2;
while k <= n loop
M[n] := M[n] - M[n/k];
k := k + 1;
end loop;
n := n + 1;
end loop;
# Find zeroes and crossings
var zero: uint8 := 0;
var cross: uint8 := 0;
n := 1;
while n < @sizeof M loop
if M[n] == 0 then
zero := zero + 1;
if M[n-1] != 0 then
cross := cross + 1;
end if;
end if;
n := n + 1;
end loop;
# Print table
print("The first 99 Mertens numbers are:\n");
print(" ");
n := 1;
var col: uint8 := 9;
while n < 100 loop
printtab(M[n]);
col := col - 1;
if col == 0 then
print_nl();
col := 10;
end if;
n := n + 1;
end loop;
print("M(n) is zero "); print_i8(zero); print(" times\n");
print("M(n) crosses zero "); print_i8(cross); print(" times\n");

95
Task/Mertens-function/Delphi/mertens-function.delphi Normal file
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program Mertens_function;
{$APPTYPE CONSOLE}
uses
System.SysUtils;
type
TMertens = record
merts: TArray<Integer>;
zeros, crosses: Integer;
class function Mertens(_to: Integer): TMertens; static;
end;
{ TMertens }
class function TMertens.Mertens(_to: Integer): TMertens;
var
sum, zeros, crosses: Integer;
begin
if _to < 1 then
_to := 1;
sum := 0;
zeros := 0;
crosses := 0;
SetLength(Result.merts, _to + 1);
var primes := [2];
for var i := 1 to _to do
begin
var j := i;
var cp := 0;
var spf := false;
for var p in primes do
begin
if p > j then
Break;
if j mod p = 0 then
begin
j := j div p;
inc(cp);
end;
if j mod p = 0 then
begin
spf := true;
Break;
end;
end;
if (cp = 0) and (i > 2) then
begin
cp := 1;
SetLength(primes, Length(primes) + 1);
primes[High(primes)] := i;
end;
if not spf then
begin
if cp mod 2 = 0 then
inc(sum)
else
dec(sum);
end;
Result.merts[i] := sum;
if sum = 0 then
begin
inc(zeros);
if (i > 1) and (Result.merts[i - 1] <> 0) then
inc(crosses);
end;
end;
Result.zeros := zeros;
Result.crosses := crosses;
end;
begin
var m := TMertens.mertens(1000);
writeln('Mertens sequence - First 199 terms:');
for var i := 0 to 199 do
begin
if i = 0 then
begin
write(' ');
Continue;
end;
if i mod 20 = 0 then
writeln;
write(format(' %3d', [m.merts[i]]));
end;
writeln(#10#10'Equals zero ', m.zeros, ' times between 1 and 1000');
writeln(#10'Crosses zero ', m.crosses, ' times between 1 and 1000');
{$IFNDEF UNIX} readln; {$ENDIF}
end.

38
Task/Mertens-function/Draco/mertens-function.draco Normal file
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proc nonrec mertens([*] short m) void:
word n,k;
m[1] := 1;
for n from 2 upto dim(m,1)-1 do
m[n] := 1;
for k from 2 upto n do
m[n] := m[n] - m[n/k]
od
od
corp
proc nonrec main() void:
[1001] short m;
word x, y, eqz, crossz;
mertens(m);
writeln("The first 99 Mertens numbers are:");
for y from 0 by 10 upto 90 do
for x from 0 upto 9 do
if x+y=0
then write(" ")
else write(m[x+y]:3)
fi
od;
writeln()
od;
eqz := 0;
crossz := 0;
for x from 2 upto dim(m,1)-1 do
if m[x]=0 then
eqz := eqz + 1;
if m[x-1]~=0 then crossz := crossz + 1 fi
fi
od;
writeln("M(N) is zero ",eqz," times.");
writeln("M(N) crosses zero ",crossz," times.")
corp

41
Task/Mertens-function/Dyalect/mertens-function.dyalect Normal file
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func mertensNumbers(max) {
let mertens = Array.Empty(max + 1, 1)
for n in 2..max {
for k in 2..n {
mertens[n] -= mertens[n / k]
}
}
mertens
}
let max = 1000
let mertens = mertensNumbers(max)
let count = 200
let columns = 20
print("First \(count - 1) Mertens numbers:")
for i in 0..<count {
if i % columns > 0 {
print(" ", terminator: "")
}
print(i == 0 ? " " : mertens[i].ToString().PadLeft(2, ' ') + " ", terminator: "")
if (i + 1) % columns == 0 {
print()
}
}
var (zero, cross, previous) = (0, 0, 0)
for i in 1..max {
let m = mertens[i]
if m == 0 {
zero += 1
if previous != 0 {
cross += 1
}
}
previous = m
}
print("M(n) is zero \(zero) times for 1 <= n <= \(max).")
print("M(n) crosses zero \(cross) times for 1 <= n <= \(max).")

7
Task/Mertens-function/F-Sharp/mertens-function.fs Normal file
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// Mertens function. Nigel Galloway: January 31st., 2021
let mertens=mobius|>Seq.scan((+)) 0|>Seq.tail
mertens|>Seq.take 500|>Seq.chunkBySize 25|>Seq.iter(fun n->Array.iter(printf "%3d") n;printfn "\n####")
let n=mertens|>Seq.take 1000|>Seq.mapi(fun n g->(n+1,g))|>Seq.groupBy snd|>Map.ofSeq
n|>Map.iter(fun n g->printf "%3d->" n; g|>Seq.iter(fun(n,_)->printf "%3d " n); printfn "\n####")
printfn "%d Zeroes\n####" (Seq.length (snd n.[0]))
printfn "Crosses zero %d times" (mertens|>Seq.take 1000|>Seq.pairwise|>Seq.sumBy(fun(n,g)->if n<>0 && g=0 then 1 else 0)))

18
Task/Mertens-function/Factor/mertens-function.factor Normal file
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USING: formatting grouping io kernel math math.extras
math.ranges math.statistics prettyprint sequences ;
! Take the cumulative sum of the mobius sequence to avoid
! summing lower terms over and over.
: mertens-upto ( n -- seq ) [1,b] [ mobius ] map cum-sum ;
"The first 199 terms of the Mertens sequence:" print
199 mertens-upto " " prefix 20 group
[ [ "%3s" printf ] each nl ] each nl
"In the first 1,000 terms of the Mertens sequence there are:"
print 1000 mertens-upto
[ [ zero? ] count bl pprint bl "zeros." print ]
[
2 <clumps> [ first2 [ 0 = not ] [ zero? ] bi* and ] count bl
pprint bl "zero crossings." print
] bi

51
Task/Mertens-function/Forth/mertens-function.fth Normal file
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: AMOUNT 1000 ;
variable mertens AMOUNT cells allot
: M 1- cells mertens + ; \ 1-indexed array
: make-mertens
1 1 M !
2 begin dup AMOUNT <= while
1 over M !
2 begin over over >= while
over over / M @
2 pick M @ swap -
2 pick M !
1+ repeat
drop
1+ repeat
drop
;
: print-row
begin dup while
swap dup M @ 3 .r 1+
swap 1-
repeat
drop
;
: print-table ." "
1 9 print-row cr
begin dup 100 < while 10 print-row cr repeat
drop
;
: find-zero-cross
0 0
1 begin dup AMOUNT <= while
dup M @ 0= if
swap 1+ swap
dup 1- M @ 0<> if rot 1+ -rot then
then
1+
repeat
drop
;
make-mertens
." The first 99 Mertens numbers are:" cr print-table
find-zero-cross
." M(N) is zero " . ." times." cr
." M(N) crosses zero " . ." times." cr
bye

40
Task/Mertens-function/Fortran/mertens-function.f Normal file
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program Mertens
implicit none
integer M(1000), n, k, zero, cross
C Generate Mertens numbers
M(1) = 1
do 10 n=2, 1000
M(n) = 1
do 10 k=2, n
M(n) = M(n) - M(n/k)
10 continue
C Print table
write (*,"('The first 99 Mertens numbers are:')")
write (*,"(' ')",advance='no')
k = 9
do 20 n=1, 99
write (*,'(I3)',advance='no') M(n)
k = k-1
if (k .EQ. 0) then
k=10
write (*,*)
end if
20 continue
C Calculate zeroes and crossings
zero = 0
cross = 0
do 30 n=2, 1000
if (M(n) .EQ. 0) then
zero = zero + 1
if (M(n-1) .NE. 0) cross = cross+1
end if
30 continue
40 format("M(N) is zero ",I2," times.")
write (*,40) zero
50 format("M(N) crosses zero ",I2," times.")
write (*,50) cross
end program

34
Task/Mertens-function/FreeBASIC/mertens-function.basic Normal file
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function padto( i as ubyte, j as integer ) as string
return wspace(i-len(str(j)))+str(j)
end function
dim as integer M( 1 to 1000 ), n, col, k, psum
dim as integer num_zeroes = 0, num_cross = 0
dim as string outstr
M(1) = 1
for n = 2 to 1000
psum = 0
for k = 2 to n
psum += M(int(n/k))
next k
M(n) = 1 - psum
if M(n) = 0 then
num_zeroes += 1
if M(n-1)<>0 then
num_cross += 1
end if
end if
next n
print using "There are ### zeroes in the range 1 to 1000."; num_zeroes
print using "There are ### crossings in the range 1 to 1000."; num_cross
print "The first 100 Mertens numbers are: "
for n=1 to 100
outstr += padto(3, M(n))+" "
if n mod 10 = 0 then
print outstr
outstr = ""
end if
next n

66
Task/Mertens-function/Go/mertens-function.go Normal file
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package main
import "fmt"
func mertens(to int) ([]int, int, int) {
if to < 1 {
to = 1
}
merts := make([]int, to+1)
primes := []int{2}
var sum, zeros, crosses int
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 {
sum++
} else {
sum--
}
}
merts[i] = sum
if sum == 0 {
zeros++
if i > 1 && merts[i-1] != 0 {
crosses++
}
}
}
return merts, zeros, crosses
}
func main() {
merts, zeros, crosses := mertens(1000)
fmt.Println("Mertens 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", merts[i])
}
fmt.Println("\n\nEquals zero", zeros, "times between 1 and 1000")
fmt.Println("\nCrosses zero", crosses, "times between 1 and 1000")
}

34
Task/Mertens-function/Haskell/mertens-function.hs Normal file
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import Data.List.Split (chunksOf)
import qualified Data.MemoCombinators as Memo
import Math.NumberTheory.Primes (unPrime, factorise)
import Text.Printf (printf)
moebius :: Integer -> Int
moebius = product . fmap m . factorise
where
m (p, e)
| unPrime p == 0 = 0
| e == 1 = -1
| otherwise = 0
mertens :: Integer -> Int
mertens = Memo.integral (\n -> sum $ fmap moebius [1..n])
countZeros :: [Integer] -> Int
countZeros = length . filter ((==0) . mertens)
crossesZero :: [Integer] -> Int
crossesZero = length . go . fmap mertens
where
go (x:y:xs)
| y == 0 && x /= 0 = y : go (y:xs)
| otherwise = go (y:xs)
go _ = []
main :: IO ()
main = do
printf "The first 99 terms for M(1..99):\n\n "
mapM_ (printf "%3d" . mertens) [1..9] >> printf "\n"
mapM_ (\row -> mapM_ (printf "%3d" . mertens) row >> printf "\n") $ chunksOf 10 [10..99]
printf "\nM(n) is zero %d times for 1 <= n <= 1000.\n" $ countZeros [1..1000]
printf "M(n) crosses zero %d times for 1 <= n <= 1000.\n" $ crossesZero [1..1000]

12
Task/Mertens-function/J/mertens-function.j Normal file
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mu =: 0:`(1 - 2 * 2|#@{.)@.(1: = */@{:)@(2&p:)"0
M =: +/@([: mu 1:+i.)
m1000 =: (M"0) 1+i.1000
zero =: +/ m1000 = 0
cross =: +/ (-.*.1:|]) m1000 ~: 0
echo 'The first 99 Merten numbers are'
echo 10 10$ __, 99{.m1000
echo 'M(N) is zero ',(":zero),' times.'
echo 'M(N) crosses zero ',(":cross),' times.'
exit''

109
Task/Mertens-function/Java/mertens-function.java Normal file
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public class MertensFunction {
public static void main(String[] args) {
System.out.printf("First 199 terms of the merten function are as follows:%n ");
for ( int n = 1 ; n < 200 ; n++ ) {
System.out.printf("%2d ", mertenFunction(n));
if ( (n+1) % 20 == 0 ) {
System.out.printf("%n");
}
}
for ( int exponent = 3 ; exponent<= 8 ; exponent++ ) {
int zeroCount = 0;
int zeroCrossingCount = 0;
int positiveCount = 0;
int negativeCount = 0;
int mSum = 0;
int mMin = Integer.MAX_VALUE;
int mMinIndex = 0;
int mMax = Integer.MIN_VALUE;
int mMaxIndex = 0;
int nMax = (int) Math.pow(10, exponent);
for ( int n = 1 ; n <= nMax ; n++ ) {
int m = mertenFunction(n);
mSum += m;
if ( m < mMin ) {
mMin = m;
mMinIndex = n;
}
if ( m > mMax ) {
mMax = m;
mMaxIndex = n;
}
if ( m > 0 ) {
positiveCount++;
}
if ( m < 0 ) {
negativeCount++;
}
if ( m == 0 ) {
zeroCount++;
}
if ( m == 0 && mertenFunction(n - 1) != 0 ) {
zeroCrossingCount++;
}
}
System.out.printf("%nFor M(x) with x from 1 to %,d%n", nMax);
System.out.printf("The maximum of M(x) is M(%,d) = %,d.%n", mMaxIndex, mMax);
System.out.printf("The minimum of M(x) is M(%,d) = %,d.%n", mMinIndex, mMin);
System.out.printf("The sum of M(x) is %,d.%n", mSum);
System.out.printf("The count of positive M(x) is %,d, count of negative M(x) is %,d.%n", positiveCount, negativeCount);
System.out.printf("M(x) has %,d zeroes in the interval.%n", zeroCount);
System.out.printf("M(x) has %,d crossings in the interval.%n", zeroCrossingCount);
}
}
private static int MU_MAX = 100_000_000;
private static int[] MU = null;
private static int[] MERTEN = null;
// Compute mobius and merten function via sieve
private static int mertenFunction(int n) {
if ( MERTEN != null ) {
return MERTEN[n];
}
// Populate array
MU = new int[MU_MAX+1];
MERTEN = new int[MU_MAX+1];
MERTEN[1] = 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;
}
}
}
int sum = 1;
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;
}
sum += MU[i];
MERTEN[i] = sum;
}
return MERTEN[n];
}
}

19
Task/Mertens-function/Jq/mertens-function-1.jq Normal file
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def sum(s): reduce s as $x (null; . + $x);
def nwise($n):
def n: if length <= $n then . else .[0:$n] , (.[$n:] | n) end;
n;
def lpad($len): tostring | ($len - length) as $l | (" " * $l)[:$l] + .;
# input: an array
# output: number of crossings at $value
def count_crossings($value):
. as $a
| reduce range(0; length) as $i ({};
if $a[$i] == $value
then if $i == 0 or .prev != $value then .count += 1 else . end
else .
end
| .prev = $a[$i] )
| .count;

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Task/Mertens-function/Jq/mertens-function-2.jq Normal file
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# Input: $max >= 1
# Output: an array of size $max with $max mertenNumbers beginning with 1
def mertensNumbers:
. as $max
| reduce range(2; $max + 1) as $n ( [1];
.[$n-1]=1
| reduce range(2; $n+1) as $k (.;
.[$n-1] -= .[($n / $k) | floor - 1] ));

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Task/Mertens-function/Jq/mertens-function-3.jq Normal file
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# Task 0:
def mertens_number:
mertensNumbers[.-1];
def task1:
"The first \(.) Mertens numbers are:",
(mertensNumbers | nwise(10) | map(lpad(2)) | join(" ") );
def task2:
. as $n
| sum(mertensNumbers[] | select(.==0) | 1)
| "M(n) is zero \(.) times for 1 <= n <= \($n)\n";
def task3:
. as $n
| mertensNumbers
| count_crossings(0)
| "M(n) crosses zero \(.) times for 1 <= n <= \($n).\n" ;
(99|task1),
"",
(1000 | (task2, task3))

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Task/Mertens-function/Julia/mertens-function.julia Normal file
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using Primes, Formatting
function moebius(n::Integer)
@assert n > 0
m(p, e) = p == 0 ? 0 : e == 1 ? -1 : 0
return reduce(*, m(p, e) for (p, e) in factor(n) if p ≥ 0; init=1)
end
μ(n) = moebius(n)
mertens(x) = sum(n -> μ(n), 1:x)
M(x) = mertens(x)
print("First 99 terms of the Mertens function for positive integers:\n ")
for n in 1:99
print(lpad(M(n), 3), n % 10 == 9 ? "\n" : "")
end
function maximinM(N)
z, cros, lastM, maxi, maxM, mini, minM, sumM, pos, neg = 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
for i in 1:N
m = μ(i) + lastM
if m == 0 && lastM != 0
cros += 1
end
sumM += m
lastM = m
if m > maxM
maxi = i
maxM = m
elseif m < minM
mini = i
minM = m
end
if m > 0
pos += 1
elseif m < 0
neg += 1
else
z += 1
end
end
println("\nFor M(x) with x from 1 to $(format(N, commas=true)):")
println("The maximum of M(x) is M($(format(maxi, commas=true)) = $maxM.")
println("The minimum of M(x) is M($(format(mini, commas=true))) = $minM.")
println("The sum of M(x) is $(format(sumM, commas=true)).")
println("The count of positive M(x) is $(format(pos, commas=true)), count of negative M(x) is $(format(neg, commas=true)).")
println("M(x) has $(format(z, commas=true)) zeroes in the interval.")
println("M(x) has $(format(cros, commas=true)) crossings in the interval.")
diff = pos - neg
if diff > 0
println("Positive M(x) exceed negative ones by $(format(diff, commas=true)).")
else
println("Negative M(x) exceed positive ones by $(format(-diff, commas=true)).")
end
end
foreach(maximinM, (1000, 1_000_000, 1_000_000_000))

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Task/Mertens-function/MAD/mertens-function.mad Normal file
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NORMAL MODE IS INTEGER
DIMENSION M(1000)
M(1) = 1
THROUGH GENMRT, FOR N=2, 1, N.G.1000
M(N) = 1
THROUGH GENMRT, FOR K=2, 1, K.G.N
GENMRT M(N) = M(N) - M(N/K)
PRINT COMMENT $ FIRST 99 MERTEN NUMBERS ARE$
VECTOR VALUES F9 = $S3,9(I2,S1)*$
VECTOR VALUES F10 = $10(I2,S1)*$
PRINT FORMAT F9, M(1), M(2), M(3), M(4), M(5), M(6),
0 M(7), M(8), M(9)
THROUGH SHOW, FOR N=10, 10, N.GE.100
SHOW PRINT FORMAT F10, M(N), M(N+1), M(N+2), M(N+3), M(N+4),
0 M(N+5), M(N+6), M(N+7), M(N+8), M(N+9), M(N+10)
ZERO = 0
CROSS = 0
THROUGH ZC, FOR N=1, 1, N.G.1000
WHENEVER M(N).E.0, ZERO = ZERO + 1
ZC WHENEVER M(N).E.0 .AND. M(N-1).NE.0, CROSS = CROSS + 1
VECTOR VALUES FZ = $13HM(N) IS ZERO ,I2,S1,5HTIMES*$
PRINT FORMAT FZ, ZERO
VECTOR VALUES FC = $18HM(N) CROSSES ZERO ,I2,S1,5HTIMES*$
PRINT FORMAT FC, CROSS
END OF PROGRAM

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Task/Mertens-function/Mathematica/mertens-function.math Normal file
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ClearAll[Mertens]
Mertens[n_] := Total[MoebiusMu[Range[n]]]
Grid[Partition[Mertens /@ Range[99], UpTo[10]]]
Count[Mertens /@ Range[1000], 0]
SequenceCount[Mertens /@ Range[1000], {Except[0], 0}]

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Task/Mertens-function/Modula-2/mertens-function.mod2 Normal file
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MODULE Mertens;
FROM InOut IMPORT WriteString, WriteInt, WriteCard, WriteLn;
CONST Max = 1000;
VAR n, k, x, y, zero, cross: CARDINAL;
M: ARRAY [1..Max] OF INTEGER;
BEGIN
M[1] := 1;
FOR n := 2 TO Max DO
M[n] := 1;
FOR k := 2 TO n DO
M[n] := M[n] - M[n DIV k];
END;
END;
WriteString("The first 99 Mertens numbers are:");
WriteLn();
FOR y := 0 TO 90 BY 10 DO
FOR x := 0 TO 9 DO
IF x+y=0 THEN WriteString(" ");
ELSE WriteInt(M[x+y], 3);
END;
END;
WriteLn();
END;
zero := 0;
cross := 0;
FOR n := 2 TO Max DO
IF M[n] = 0 THEN
zero := zero + 1;
IF M[n-1] # 0 THEN
cross := cross + 1;
END;
END;
END;
WriteString("M(n) is zero ");
WriteCard(zero,0);
WriteString(" times.");
WriteLn();
WriteString("M(n) crosses zero ");
WriteCard(cross,0);
WriteString(" times.");
WriteLn();
END Mertens.

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Task/Mertens-function/Nim/mertens-function.nim Normal file
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import sequtils, strformat
func mertensNumbers(max: int): seq[int] =
result = repeat(1, max + 1)
for n in 2..max:
for k in 2..n:
dec result[n], result[n div k]
const Max = 1000
let mertens = mertensNumbers(Max)
echo "First 199 Mertens numbers:"
const Count = 200
var column = 0
for i in 0..<Count:
if column > 0: stdout.write ' '
stdout.write if i == 0: " " else: &"{mertens[i]:>2}"
inc column
if column == 20:
stdout.write '\n'
column = 0
var zero, cross, previous = 0
for i in 1..Max:
let m = mertens[i]
if m == 0:
inc zero
if previous != 0:
inc cross
previous = m
echo ""
echo &"M(n) is zero {zero} times for 1 ⩽ n ⩽ {Max}."
echo &"M(n) crosses zero {cross} times for 1 ⩽ n ⩽ {Max}."

35
Task/Mertens-function/PL-I/mertens-function.pli Normal file
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mertens: procedure options(main);
%replace MAX by 1000;
declare M(1:MAX) fixed binary(5);
declare (n, k) fixed binary(10);
declare (isZero, crossZero) fixed binary(8);
M(1) = 1;
do n = 2 to MAX;
M(n) = 1;
do k = 2 to n;
M(n) = M(n) - M(divide(n,k,10));
end;
end;
put skip list('The first 99 Mertens numbers are:');
put skip list(' ');
do n = 1 to 99;
put edit(M(n)) (F(3));
if mod(n,10) = 9 then put skip;
end;
isZero = 0;
crossZero = 0;
do n = 2 to MAX;
if M(n) = 0 then do;
isZero = isZero + 1;
if M(n-1) ^= 0 then
crossZero = crossZero + 1;
end;
end;
put skip list('Zeroes: ',isZero);
put skip list('Crossings:',crossZero);
end mertens;

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Task/Mertens-function/Pascal/mertens-function.pas Normal file
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program Merten;
{$IFDEF FPC}
{$MODE DELPHI}
{$Optimization ON,ALL}
{$ELSE}
{$APPTYPE CONSOLE}
{$ENDIF}
uses
sysutils;
const
BigLimit = 10*1000*1000*1000;//1e10
type
tSieveElement = Int8;
tpSieve = pInt8;
tMoebVal = array[-1..1] of Int64;
var
MertensValues : array[-40000..50500] of NativeInt;
primes : array of byte;
sieve : array of tSieveElement;
procedure CompactPrimes;
//searching for needed primes
//last primes are marked with -1
var
pSieve : tpSieve;
i,lmt,dp:NativeInt;
Begin
setlength(Primes,74500);//suffices for primes to calc square upto 1e12
//extract difference of primes
i := 2;
lmt := 0;
dp := 2;
pSieve :=@sieve[0];
repeat
IF pSieve[i]= 0 then
Begin
//mark for Moebius
pSieve[i]:= -1;
primes[lmt] := dp;
dp := 0;
inc(lmt);
end;
inc(dp);
inc(i);
until i*i >BigLimit;
setlength(Primes,lmt+1);
repeat
IF pSieve[i]= 0 then
//mark for Moebius
pSieve[i]:= -1;
inc(i);
until i >BigLimit;
end;
procedure SieveSquares;
//mark all powers >=2 of prime => all powers = 2 is sufficient
var
pSieve : tpSieve;
i,sq,k,prime : NativeInt;
Begin
pSieve := @sieve[0];
prime := 0;
For i := 0 to High(primes) do
Begin
prime := prime+primes[i];
sq := prime*prime;
k := sq;
if sq > BigLimit then
break;
repeat
pSieve[k] := 0;
inc(k,sq);
until k> BigLimit;
end;
end;
procedure initPrimes;
var
pSieve : tpSieve;
fakt,
sieveprime : NativeUint;
begin
pSieve := @sieve[0];
sieveprime := 2;
repeat
if pSieve[sieveprime]=0 then
begin
fakt := sieveprime+sieveprime;
while fakt <=BigLimit do
Begin
//count divisors
inc(pSieve[fakt]);
inc(fakt,sieveprime);
end;
end;
inc(sieveprime);
until sieveprime>BigLimit DIV 2;
//Möbius of 1
pSieve[1] := 1;
//convert to Moebius
For fakt := 2 to BigLimit do
Begin
sieveprime := pSieve[fakt];
IF sieveprime<>0 then
pSieve[fakt] := 1-(2*(sieveprime AND 1)) ;
end;
CompactPrimes;
SieveSquares;
end;
procedure OutMerten10(Lmt,ZeroCross:NativeInt;Const MoebVal:tMoebVal);
var
i,j: NativeInt;
Begin
Writeln(lmt:11,MoebVal[-1]:11,MoebVal[1]:11,MoebVal[-1]+MoebVal[1]:11,
MoebVal[-1]-MoebVal[1]:7,MoebVal[0]:11);
i:= low(MertensValues);
while MertensValues[i] = 0 do
inc(i);
j:= High(MertensValues);
while MertensValues[j] = 0 do
dec(j);
write('Merten min ',i:6,' max ',j:6,' zero''s ',MertensValues[0]:8);
writeln(' zeroCross ',ZeroCross);
writeln;
end;
procedure Count_x10;
var
MoebCount: tMoebVal;
pSieve : tpSieve;
i,lmt,Merten,Moebius,LastMert,ZeroCross: NativeInt;
begin
writeln('[1 to limit]');
Writeln('Limit Moeb. odd Moeb.even sqr-free Merten Zero''s');
pSieve := @sieve[0];
For i := -1 to 1 do
MoebCount[i]:=0;
ZeroCross := 0;
LastMert :=1;
Merten :=0;
lmt := 10;
i := 1;
repeat
while i <= lmt do
Begin
Moebius := pSieve[i];
inc(MoebCount[Moebius]);
inc(Merten,Moebius);
inc(MertensValues[Merten]);//MoebCount[1]-MoebCount[-1]]);
inc(ZeroCross,ORD( (Merten = 0) AND (LastMert <> 0)));
LastMert := Merten;
inc(i);
end;
OutMerten10(Lmt,ZeroCross,MoebCount);
IF lmt >= BigLimit then
BREAK;
lmt := lmt*10;
IF lmt >BigLimit then
lmt := BigLimit;
until false;
writeln;
end;
procedure OutMerten(lmt:NativeInt);
var
i,k,m : NativeInt;
Begin
iF lmt> BigLimit then
lmt := BigLimit;
writeln('Mertens numbers from 1 to ',lmt);
k := 9;
write('':3);
m := 0;
For i := 1 to lmt do
Begin
inc(m,sieve[i]);
write(m:3);
dec(k);
IF k = 0 then
Begin
writeln;
k := 10;
end;
end;
writeln;
end;
procedure OutMoebius(lmt:NativeInt);
var
i,k : NativeInt;
Begin
iF lmt> BigLimit then
lmt := BigLimit;
writeln('Möbius numbers from 1 to ',lmt);
k := 19;
write('':3);
For i := 1 to lmt do
Begin
write(sieve[i]:3);
dec(k);
IF k = 0 then
Begin
writeln;
k := 20;
end;
end;
writeln;
end;
Begin
setlength(sieve,BigLimit+1);
InitPrimes;
SieveSquares;
Count_x10;
OutMoebius(199);
OutMerten(99);
setlength(primes,0);
setlength(sieve,0);
end.

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Task/Mertens-function/Perl/mertens-function.pl Normal file
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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
}
sub progressive_sum {
my @sum = shift @_;
push @sum, $sum[-1] + $_ for @_;
@sum
}
my($upto, $show, @möebius) = (1000, 199, ());
push @möebius, μ($_) for 1..$upto;
my @mertens = progressive_sum @möebius;
say "Mertens sequence - First $show terms:\n" .
(' 'x4 . sprintf "@{['%4d' x $show]}", @mertens[0..$show-1]) =~ s/((.){80})/$1\n/gr .
sprintf("\nEquals zero %3d times between 1 and $upto", scalar grep { ! $_ } @mertens) .
sprintf "\nCrosses zero%3d times between 1 and $upto", scalar grep { ! $mertens[$_-1] and $mertens[$_] } 1 .. @mertens;

33
Task/Mertens-function/Phix/mertens-function.phix Normal file
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(phixonline)-->
<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>
<span style="color: #7060A8;">requires</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"1.0.2"</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">-- (skip arg added to join_by)</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">mcache</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">1</span><span style="color: #0000FF;">}</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">Mertens</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">m</span><span style="color: #0000FF;">=</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">mcache</span><span style="color: #0000FF;">)+</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">n</span> <span style="color: #008080;">do</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">mm</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">k</span><span style="color: #0000FF;">=</span><span style="color: #000000;">2</span> <span style="color: #008080;">to</span> <span style="color: #000000;">m</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">mm</span> <span style="color: #0000FF;">-=</span> <span style="color: #000000;">mcache</span><span style="color: #0000FF;">[</span><span style="color: #7060A8;">floor</span><span style="color: #0000FF;">(</span><span style="color: #000000;">m</span><span style="color: #0000FF;">/</span><span style="color: #000000;">k</span><span style="color: #0000FF;">)]</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #000000;">mcache</span> <span style="color: #0000FF;">&=</span> <span style="color: #000000;">mm</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">mcache</span><span style="color: #0000FF;">[</span><span style="color: #000000;">n</span><span style="color: #0000FF;">]</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">constant</span> <span style="color: #000000;">first</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">99</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">perline</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">10</span>
<span style="color: #000080;font-style:italic;">--constant first = 199, perline = 20 -- matches C/Go/etc
--constant first = 143, perline = 12 -- matches wp</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #008000;">" ."</span><span style="color: #0000FF;">}&</span><span style="color: #7060A8;">apply</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">tagset</span><span style="color: #0000FF;">(</span><span style="color: #000000;">first</span><span style="color: #0000FF;">),</span><span style="color: #000000;">Mertens</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"First %d Mertens numbers:\n"</span><span style="color: #0000FF;">,</span><span style="color: #000000;">first</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">puts</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">join_by</span><span style="color: #0000FF;">(</span><span style="color: #000000;">s</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">perline</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" "</span><span style="color: #0000FF;">,</span><span style="color: #000000;">fmt</span><span style="color: #0000FF;">:=</span><span style="color: #008000;">"%3d"</span><span style="color: #0000FF;">,</span><span style="color: #000000;">skip</span><span style="color: #0000FF;">:=</span><span style="color: #000000;">1</span><span style="color: #0000FF;">))</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">prev</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">zeroes</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">crosses</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">=</span><span style="color: #000000;">2</span> <span style="color: #008080;">to</span> <span style="color: #000000;">1000</span> <span style="color: #008080;">do</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">m</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">Mertens</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">m</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">zeroes</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">1</span>
<span style="color: #000000;">crosses</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">prev</span><span style="color: #0000FF;">!=</span><span style="color: #000000;">0</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #000000;">prev</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">m</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"\nMertens[1..1000] equals zero %d times and crosses zero %d times\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">zeroes</span><span style="color: #0000FF;">,</span><span style="color: #000000;">crosses</span><span style="color: #0000FF;">})</span>
<!--

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Task/Mertens-function/Prolog/mertens-function.pro Normal file
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:- dynamic mertens_number_cache/2.
mertens_number(1, 1):- !.
mertens_number(N, M):-
mertens_number_cache(N, M),
!.
mertens_number(N, M):-
N >= 2,
mertens_number(N, 2, M, 0),
assertz(mertens_number_cache(N, M)).
mertens_number(N, N, M, M):- !.
mertens_number(N, K, M, S):-
N1 is N // K,
mertens_number(N1, M1),
K1 is K + 1,
S1 is S - M1,
mertens_number(N, K1, M, S1).
print_mertens_numbers(Count):-
print_mertens_numbers(Count, 0).
print_mertens_numbers(Count, Count):-!.
print_mertens_numbers(Count, N):-
(N == 0 ->
write(' ')
;
mertens_number(N, M),
writef('%3r', [M])
),
N1 is N + 1,
Column is N1 mod 20,
(N > 0, Column == 0 ->
nl
;
true
),
print_mertens_numbers(Count, N1).
count_zeros(From, To, Z, C):-
count_zeros(From, To, Z, C, 0, 0, 0).
count_zeros(From, To, Z, C, Z, C, _):-
From > To,
!.
count_zeros(From, To, Z, C, Z1, C1, P):-
mertens_number(From, M),
(M == 0 -> Z2 is Z1 + 1 ; Z2 = Z1),
(M == 0, P \= 0 -> C2 is C1 + 1 ; C2 = C1),
Next is From + 1,
count_zeros(Next, To, Z, C, Z2, C2, M).
main:-
writeln('First 199 Mertens numbers:'),
print_mertens_numbers(200),
count_zeros(1, 1000, Z, C),
writef('M(n) is zero %t times for 1 <= n <= 1000.\n', [Z]),
writef('M(n) crosses zero %t times for 1 <= n <= 1000.\n', [C]).

14
Task/Mertens-function/PureBasic/mertens-function.basic Normal file
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Dim M.i(1000)
M(1)=1
For n=2 To 1000
psum=0
For k=2 To n : psum+M(Int(n/k)) : Next : M(n)=1-psum
If M(n)=0 : z+1 : If M(n-1)<>0 : c+1 : EndIf : EndIf
Next
OpenConsole("")
PrintN("First 99 Mertens numbers:") : Print(Space(4))
For n=1 To 99 : Print(RSet(Str(M(n)),4)) : If n%10=9 : PrintN("") : EndIf : Next
PrintN("M(N) is zero "+Str(z)+" times.") : PrintN("M(N) crosses zero "+Str(c)+" times.")
Input()

26
Task/Mertens-function/Python/mertens-function.py Normal file
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def mertens(count):
"""Generate Mertens numbers"""
m = [None, 1]
for n in range(2, count+1):
m.append(1)
for k in range(2, n+1):
m[n] -= m[n//k]
return m
ms = mertens(1000)
print("The first 99 Mertens numbers are:")
print(" ", end=' ')
col = 1
for n in ms[1:100]:
print("{:2d}".format(n), end=' ')
col += 1
if col == 10:
print()
col = 0
zeroes = sum(x==0 for x in ms)
crosses = sum(a!=0 and b==0 for a,b in zip(ms, ms[1:]))
print("M(N) equals zero {} times.".format(zeroes))
print("M(N) crosses zero {} times.".format(crosses))

37
Task/Mertens-function/Quackery/mertens-function.quackery Normal file
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[ ' [ 0 ]
swap 1+ times
[ dup -1 peek
i^ 1+ mobius
+ join ]
behead drop ] is mertens ( n --> [ )
[ say " "
99 times
[ dup i^ peek
dup dup
-1 > if sp
abs 10 < if sp
echo
i^ 1+ 10 mod
9 = if cr ]
drop ] is grid ( [ --> )
[ 0 swap
witheach
[ 0 = + ] ] is zeroes ( [ --> n )
[ 0 0
rot witheach
[ dup 0 =
rot 0 !=
and
rot + swap ]
drop ] is crossings ( [ --> n )
1000 mertens
say "First 99 terms:"
cr
dup grid
cr
dup zeroes echo say " zeroes and "
crossings echo say " crossings"

61
Task/Mertens-function/REXX/mertens-function.rexx Normal file
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/*REXX pgm computes & shows a value grid of the Mertens function for a range of integers*/
parse arg LO HI grp eqZ xZ . /*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 /* " " " " " " */
if eqZ=='' | eqZ=="," then eqZ= 1000 /* " " " " " " */
if xZ=='' | xZ=="," then xZ= 1000 /* " " " " " " */
call genP /*generate primes up to max √ HIHI */
call Franz LO, HI
if eqZ>0 then call Franz 1, -eqZ
if xZ>0 then call Franz -1, xZ
exit 0 /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
Franz: parse arg a 1 oa,b 1 ob; @Mertens= ' The Mertens sequence from '
a= abs(a); b= abs(b); grid= oa>=0 & ob>=0 /*semaphore used to show a grid title. */
if grid then say center(@Mertens LO " ──► " HI" ", max(50, grp*3), '═') /*show title*/
else say
zeros= 0 /*# of 0's found for Mertens function.*/
Xzero= 0 /*number of times that zero was crossed*/
$=; prev= /*$ holds output grid of GRP numbers. */
do j=a to b; _= Mertens(j) /*process some numbers from LO ──► HI.*/
if _==0 then zeros= zeros + 1 /*Is Zero? Then bump the zeros counter*/
if _==0 then if prev\==0 then Xzero= Xzero+1 /*prev ¬=0? " " " Xzero " */
prev= _
if grid then $= $ right(_, 2) /*build grid if A & B are non─negative.*/
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*/
if oa<0 then say @Mertens a " to " b ' has crossed zero ' Xzero " times."
if ob<0 then say @Mertens a " to " b ' has ' zeros " zeros."
return
/*──────────────────────────────────────────────────────────────────────────────────────*/
Mertens: procedure expose @. !!. M.; parse arg n; if M.n\==. then return M.n
if n<1 then return '∙'; m= 0 /*handle special cases of non─positive#*/
do k=1 for n; m= m + mobius(k) /*sum the MU's up to N. */
end /*k*/ /* [↑] mobius function uses memoization*/
M.n= m; return m /*return the sum of all the MU's. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
mobius: procedure expose @. !!.; parse arg x 1 ox /*get integer to be tested for mu */
if !!.x\==. then return !!.x /*X computed before? Return that value*/
if x<1 then return '∙'; mu= 0 /*handle special case of non-positive #*/
do k=1; p= @.k; if p>x then leave /* (P) > X? Then we're done.*/
if p*p>x then do; mu= mu+1; leave; end /* (P**2) > X? Bump # and leave*/
if x//p==0 then do; mu= mu+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*/ /*MU (below) is almost always small, <9*/
!!.ox= -1 ** mu; return !!.ox /*raise -1 to the mu power, memoize it.*/
/*──────────────────────────────────────────────────────────────────────────────────────*/
genP: @.1=2; @.2=3; @.3=5; @.4=7; @.5=11; @.6=13 /*initialize some low primes; # primes.*/
!!.=.; M.=!!.; #= 6; sq.#= @.6**2 /* " 2 arrays for memoization. */
do j=@.#+4 by 2 to max(HI, eqZ, xZ); parse var j '' -1 _ /*odd Ps from now on*/
if _==5 then iterate; if j//3==0 then iterate; if j//7==0 then iterate /*÷ 5 3 7*/
do k=7 while sq.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. */
#= #+1; @.#=j; sq.j= j*j /*bump P count; P──►@.; compute J**2*/
end /*j*/; return /*calculate the squares of some primes.*/

21
Task/Mertens-function/Raku/mertens-function.raku Normal file
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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 @mertens = lazy [\+] flat '', 1, (2..*).hyper.map: -> \n { μ(n) };
put "Mertens sequence - First 199 terms:\n",
@mertens[^200]».fmt('%3s').batch(20).join("\n"),
"\n\nEquals zero ", +@mertens[1..1000].grep( !* ),
' times between 1 and 1000', "\n\nCrosses zero ",
+@mertens[1..1000].kv.grep( {!$^v and @mertens[$^k]} ),
" times between 1 and 1000\n\nFirst Mertens equal to:";
for 10, 20, 30 … 100 -> $threshold {
printf "%4d: M(%d)\n", -$threshold, @mertens.first: * == -$threshold, :k;
printf "%4d: M(%d)\n", $threshold, @mertens.first: * == $threshold, :k;
}

18
Task/Mertens-function/Ruby/mertens-function.rb Normal file
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require 'prime'
def μ(n)
return 1 if self == 1
pd = n.prime_division
return 0 unless pd.map(&:last).all?(1)
pd.size.even? ? 1 : -1
end
def M(n)
(1..n).sum{|n| μ(n)}
end
([" "] + (1..199).map{|n|"%2s" % M(n)}).each_slice(20){|line| puts line.join(" ") }
ar = (1..1000).map{|n| M(n)}
puts "\nThe Mertens function is zero #{ar.count(0)} times in the range (1..1000);"
puts "it crosses zero #{ar.each_cons(2).count{|m1, m2| m1 != 0 && m2 == 0}} times."

25
Task/Mertens-function/Run-BASIC/mertens-function.basic Normal file
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dim M(1000)
M(1) = 1
for n = 2 to 1000
M(n) = 1
for k = 2 to n
M(n) = M(n)-M(int(n/k))
next k
next n
print "First 99 Mertens numbers:"
print " ";
for n = 1 to 99
print using("###", M(n));
if n mod 10 = 9 then print
next n
numCruza = 0
numEsCero = 0
for n = 1 to 1000
if M(n) = 0 then
numEsCero = numEsCero +1
if M(n-1) <> 0 then numCruza = numCruza +1
end if
next n
print
print "M(n) is zero "; numEsCero; " times."
print "M(n) crosses zero "; numCruza; " times."

2
Task/Mertens-function/Sidef/mertens-function-1.sidef Normal file
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say mertens(123456789) #=> 1170
say mertens(1234567890) #=> 9163

35
Task/Mertens-function/Sidef/mertens-function-2.sidef Normal file
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func mertens(n) is cached {
var lookup_size = (2 * n.iroot(3)**2)
var mertens_lookup = [0]
for k in (1..lookup_size) {
mertens_lookup[k] = (mertens_lookup[k-1] + k.moebius)
}
static cache = Hash()
func (n) {
if (n <= lookup_size) {
return mertens_lookup[n]
}
if (cache.has(n)) {
return cache{n}
}
var M = 1
var s = n.isqrt
for k in (2 .. floor(n/(s+1))) {
M -= __FUNC__(floor(n/k))
}
for k in (1..s) {
M -= (mertens_lookup[k] * (floor(n/k) - floor(n/(k+1))))
}
cache{n} = M
}(n)
}

12
Task/Mertens-function/Sidef/mertens-function-3.sidef Normal file
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with (200) {|n|
say "Mertens function in the range 1..#{n}:"
(1..n).map { mertens(_) }.slices(20).each {|line|
say line.map{ "%2s" % _ }.join(' ')
}
}
with (1000) {|n|
say "\nIn the range 1..#{n}, there are:"
say (1..n->count_by { mertens(_)==0 }, " zeros")
say (1..n->count_by { mertens(_)==0 && mertens(_-1)!=0 }, " zero crossings")
}

41
Task/Mertens-function/Swift/mertens-function.swift Normal file
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import Foundation
func mertensNumbers(max: Int) -> [Int] {
var mertens = Array(repeating: 1, count: max + 1)
for n in 2...max {
for k in 2...n {
mertens[n] -= mertens[n / k]
}
}
return mertens
}
let max = 1000
let mertens = mertensNumbers(max: max)
let count = 200
let columns = 20
print("First \(count - 1) Mertens numbers:")
for i in 0..<count {
if i % columns > 0 {
print(" ", terminator: "")
}
print(i == 0 ? " " : String(format: "%2d", mertens[i]), terminator: "")
if (i + 1) % columns == 0 {
print()
}
}
var zero = 0, cross = 0, previous = 0
for i in 1...max {
let m = mertens[i]
if m == 0 {
zero += 1
if previous != 0 {
cross += 1
}
}
previous = m
}
print("M(n) is zero \(zero) times for 1 <= n <= \(max).")
print("M(n) crosses zero \(cross) times for 1 <= n <= \(max).")

28
Task/Mertens-function/True-BASIC/mertens-function.basic Normal file
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DIM m(1000)
LET m(1) = 1
FOR n = 2 TO 1000
LET m(n) = 1
FOR k = 2 TO n
LET m(n) = m(n)-m(INT(n/k))
NEXT k
NEXT n
PRINT "First 99 Mertens numbers:"
PRINT " ";
FOR n = 1 TO 99
PRINT " ";
PRINT USING "##": m(n);
!IF REMAINDER(ROUND(n),10) = 9 THEN PRINT
IF MOD(n,10) = 9 THEN PRINT
NEXT n
LET numcruza = 0
LET numeszero = 0
FOR n = 1 TO 1000
IF m(n) = 0 THEN
LET numeszero = numeszero+1
IF m(n-1) <> 0 THEN LET numcruza = numcruza+1
END IF
NEXT n
PRINT
PRINT "M(n) is zero"; numeszero; "times."
PRINT "M(n) crosses zero"; numcruza; "times."
END

65
Task/Mertens-function/V-(Vlang)/mertens-function.v Normal file
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fn mertens(t int) ([]int, int, int) {
mut to:=t
if to < 1 {
to = 1
}
mut merts := []int{len:to+1}
mut primes := [2]
mut sum := 0
mut zeros := 0
mut crosses := 0
for i := 1; i <= to; i++ {
mut j := i
mut cp := 0 // counts prime factors
mut spf := false // true if there is a square prime factor
for p in 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 << i
}
if !spf {
if cp%2 == 0 {
sum++
} else {
sum--
}
}
merts[i] = sum
if sum == 0 {
zeros++
if i > 1 && merts[i-1] != 0 {
crosses++
}
}
}
return merts, zeros, crosses
}
fn main() {
merts, zeros, crosses := mertens(1000)
println("Mertens sequence - First 199 terms:")
for i := 0; i < 200; i++ {
if i == 0 {
print(" ")
continue
}
if i%20 == 0 {
println('')
}
print(" ${merts[i]:2}")
}
println("\n\nEquals zero $zeros times between 1 and 1000")
println("\nCrosses zero $crosses times between 1 and 1000")
}

53
Task/Mertens-function/Wren/mertens-function.wren Normal file
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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
}
var M = Fn.new { |x| (1..x).reduce { |sum, n| sum + mu.call(n) } }
System.print("The first 199 Mertens 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, M.call(i*20 + j))) ")
}
}
System.print()
}
// use the recurrence relationship for the last 2 parts rather than calling M directly
var count = 0
var mertens = M.call(1)
for (i in 2..1000) {
mertens = mertens + mu.call(i)
if (mertens == 0) count = count + 1
}
System.print("\nThe Mertens function is zero %(count) times in the range [1, 1000].")
count = 0
var prev = M.call(1)
for (i in 2..1000) {
var next = prev + mu.call(i)
if (next == 0 && prev != 0) count = count + 1
prev = next
}
System.print("\nThe Mertens function crosses zero %(count) times in the range [1, 1000].")

34
Task/Mertens-function/XBasic/mertens-function.basic Normal file
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PROGRAM "Mertens"
VERSION "0.0000"
DECLARE FUNCTION Entry ()
FUNCTION Entry ()
DIM M[1000]
M[1] = 1
FOR n = 2 TO 1000
M[n] = 1
FOR k = 2 TO n
M[n] = M[n] - M[INT(n/k)]
NEXT k
NEXT n
PRINT "First 99 Mertens numbers:"
PRINT " ";
FOR n = 1 TO 99
PRINT FORMAT$("###", M[n]);
IF n MOD 10 = 9 THEN PRINT
NEXT n
numCruza = 0
numEsCero = 0
FOR n = 1 TO 1000
IF M[n] = 0 THEN
INC numEsCero
IF M[n-1] <> 0 THEN INC numCruza
END IF
NEXT n
PRINT
PRINT "M(n) is zero"; numEsCero; " times."
PRINT "M(n) crosses zero"; numCruza; " times."
END FUNCTION
END PROGRAM

34
Task/Mertens-function/XPL0/mertens-function.xpl0 Normal file
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integer M ( 1+1000 );
integer K, Zero, Cross, N;
begin \compute values of the Mertens function
\Generate Mertens numbers
M( 1 ) := 1;
for N := 2 to 1000 do begin
M( N ) := 1;
for K := 2 to N do M( N ) := M( N ) - M( N / K )
end;
\Print table
Text(0, "The first 99 Mertens numbers are:^m^j");
Text(0, " " );
K := 9;
for N := 1 to 99 do begin
Format(3, 0);
RlOut(0, float(M(N)));
K := K - 1;
if K = 0 then begin
K := 10;
CrLf(0);
end
end;
\Calculate zeroes and crossings
Zero := 0;
Cross := 0;
for N :=2 to 1000 do begin
if M( N ) = 0 then begin
Zero := Zero + 1;
if M( N - 1 ) # 0 then Cross := Cross + 1
end
end;
Text(0, "M(N) is zero "); IntOut(0, Zero); Text(0, " times.^m^j" );
Text(0, "M(N) crosses zero "); IntOut(0, Cross); Text(0, " times.^m^j" );
end

25
Task/Mertens-function/Yabasic/mertens-function-1.basic Normal file
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dim M(1000)
M(1) = 1
for n = 2 to 1000
M(n) = 1
for k = 2 to n
M(n) = M(n) - M(int(n/k))
next k
next n
print "First 99 Mertens numbers:"
print " ";
for n = 1 to 99
print M(n) using("###");
if mod(n, 10) = 9 print
next n
numCruza = 0
numEsCero = 0
for n = 1 to 1000
if M(n) = 0 then
numEsCero = numEsCero + 1
if M(n-1) <> 0 numCruza = numCruza + 1
end if
next n
print
print "M(n) is zero ", numEsCero, " times."
print "M(n) crosses zero ", numCruza, " times."

34
Task/Mertens-function/Yabasic/mertens-function-2.basic Normal file
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#!/bin/bash
MAX=1000
m[1]=1
for n in `seq 2 $MAX`
do
m[n]=1
for k in `seq 2 $n`
do
m[n]=$((m[n]-m[n/k]))
done
done
echo 'The first 99 Mertens numbers are:'
echo -n ' '
for n in `seq 1 99`
do
printf '%2d ' ${m[n]}
test $((n%10)) -eq 9 && echo
done
zero=0
cross=0
for n in `seq 1 $MAX`
do
if [ ${m[n]} -eq 0 ]
then
((zero++))
test ${m[n-1]} -ne 0 && ((cross++))
fi
done
echo "M(N) is zero $zero times."
echo "M(N) crosses zero $cross times."

25
Task/Mertens-function/Zkl/mertens-function-1.zkl Normal file
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fcn mertensW(n){
[1..].tweak(fcn(n,pm){
pm.incN(mobius(n));
pm.value
}.fp1(Ref(0)))
}
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;
}

9
Task/Mertens-function/Zkl/mertens-function-2.zkl Normal file
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mertensW().walk(199)
.pump(Console.println, T(Void.Read,19,False),
fcn{ vm.arglist.pump(String,"%3d".fmt) });
println("\nIn the first 1,000 terms of the Mertens sequence there are:");
otm:=mertensW().pump(1_000,List);
otm.reduce(fcn(s,m){ s + (m==0) },0) : println(_," zeros");
otm.reduce(fcn(p,m,rs){ rs.incN(m==0 and p!=0); m }.fp2( s:=Ref(0) ));
println(s.value," zero crossings");