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# Tau function

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

Given a positive integer, count the number of its positive divisors.

Show the result for the first   100   positive integers.

## 11l

Translation of: Python
`F tau(n)   V ans = 0   V i = 1   V j = 1   L i * i <= n      I 0 == n % i         ans++         j = n I/ i         I j != i            ans++      i++   R ans print((1..100).map(n -> tau(n)))`
Output:
```[1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8, 2, 6, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 10, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 12, 2, 4, 6, 7, 4, 8, 2, 6, 4, 8, 2, 12, 2, 4, 6, 6, 4, 8, 2, 10, 5, 4, 2, 12, 4, 4, 4, 8, 2, 12, 4, 6, 4, 4, 4, 12, 2, 6, 6, 9]
```

## ALGOL W

Translation of: C++
`begin % find the count of the divisors of the first 100 positive integers   %    % calculates the number of divisors of v                                %    integer procedure divisor_count( integer value v ) ; begin        integer total, n, p;        total := 1; n := v;        % Deal with powers of 2 first %        while not odd( n ) do begin            total := total + 1;            n     := n div 2        end while_not_odd_n ;        % Odd prime factors up to the square root %        p := 3;        while ( p * p ) <= n do begin            integer count;            count := 1;            while n rem p = 0 do begin                count := count + 1;                n     := n div p            end while_n_rem_p_eq_0 ;            p     := p + 2;            total := total * count        end while_p_x_p_le_n ;        % If n > 1 then it's prime %        if n > 1 then total := total * 2;        total    end divisor_count ;    begin        integer limit;        limit := 100;        write( i_w := 1, s_w := 0, "Count of divisors for the first ", limit, " positive integers:" );        for n := 1 until limit do begin            if n rem 20 = 1 then write();            writeon( i_w := 3, s_w := 1, divisor_count( n ) )        end for_n    endend.`
Output:
```Count of divisors for the first 100 positive integers:
1   2   2   3   2   4   2   4   3   4   2   6   2   4   4   5   2   6   2   6
4   4   2   8   3   4   4   6   2   8   2   6   4   4   4   9   2   4   4   8
2   8   2   6   6   4   2  10   3   6   4   6   2   8   4   8   4   4   2  12
2   4   6   7   4   8   2   6   4   8   2  12   2   4   6   6   4   8   2  10
5   4   2  12   4   4   4   8   2  12   4   6   4   4   4  12   2   6   6   9
```

## AppleScript

`on factorCount(n)    if (n < 1) then return 0    set counter to 2    set sqrt to n ^ 0.5    if (sqrt mod 1 = 0) then set counter to 1    repeat with i from (sqrt div 1) to 2 by -1        if (n mod i = 0) then set counter to counter + 2    end repeat     return counterend factorCount -- Task code:local output, n, astidset output to {"Positive divisor counts for integers 1 to 100:"}repeat with n from 1 to 100    if (n mod 20 = 1) then set end of output to linefeed    set end of output to text -3 thru -1 of ("  " & factorCount(n))end repeatset astid to AppleScript's text item delimitersset AppleScript's text item delimiters to ""set output to output as textset AppleScript's text item delimiters to astidreturn output`
Output:
`"Positive divisor counts for integers 1 to 100:  1  2  2  3  2  4  2  4  3  4  2  6  2  4  4  5  2  6  2  6  4  4  2  8  3  4  4  6  2  8  2  6  4  4  4  9  2  4  4  8  2  8  2  6  6  4  2 10  3  6  4  6  2  8  4  8  4  4  2 12  2  4  6  7  4  8  2  6  4  8  2 12  2  4  6  6  4  8  2 10  5  4  2 12  4  4  4  8  2 12  4  6  4  4  4 12  2  6  6  9"`

## Arturo

`tau: function [x] -> size factors x loop split.every:20 1..100 => [    print map & => [pad to :string tau & 3]]`
Output:
```  1   2   2   3   2   4   2   4   3   4   2   6   2   4   4   5   2   6   2   6
4   4   2   8   3   4   4   6   2   8   2   6   4   4   4   9   2   4   4   8
2   8   2   6   6   4   2  10   3   6   4   6   2   8   4   8   4   4   2  12
2   4   6   7   4   8   2   6   4   8   2  12   2   4   6   6   4   8   2  10
5   4   2  12   4   4   4   8   2  12   4   6   4   4   4  12   2   6   6   9```

## AWK

` # syntax: GAWK -f TAU_FUNCTION.AWKBEGIN {    print("The tau functions for the first 100 positive integers:")    for (i=1; i<=100; i++) {      printf("%2d ",count_divisors(i))      if (i % 10 == 0) {        printf("\n")      }    }    exit(0)}function count_divisors(n,  count,i) {    for (i=1; i*i<=n; i++) {      if (n % i == 0) {        count += (i == n / i) ? 1 : 2      }    }    return(count)} `
Output:
```The tau functions for the first 100 positive integers:
1  2  2  3  2  4  2  4  3  4
2  6  2  4  4  5  2  6  2  6
4  4  2  8  3  4  4  6  2  8
2  6  4  4  4  9  2  4  4  8
2  8  2  6  6  4  2 10  3  6
4  6  2  8  4  8  4  4  2 12
2  4  6  7  4  8  2  6  4  8
2 12  2  4  6  6  4  8  2 10
5  4  2 12  4  4  4  8  2 12
4  6  4  4  4 12  2  6  6  9
```

## C

Translation of: C++
`#include <stdio.h> // See https://en.wikipedia.org/wiki/Divisor_functionunsigned int divisor_count(unsigned int n) {    unsigned int total = 1;    // Deal with powers of 2 first    for (; (n & 1) == 0; n >>= 1) {        ++total;    }    // Odd prime factors up to the square root    for (unsigned int p = 3; p * p <= n; p += 2) {        unsigned int count = 1;        for (; n % p == 0; n /= p) {            ++count;        }        total *= count;    }    // If n > 1 then it's prime    if (n > 1) {        total *= 2;    }    return total;} int main() {    const unsigned int limit = 100;    unsigned int n;     printf("Count of divisors for the first %d positive integers:\n", limit);    for (n = 1; n <= limit; ++n) {        printf("%3d", divisor_count(n));        if (n % 20 == 0) {            printf("\n");        }    }     return 0;}`
Output:
```Count of divisors for the first 100 positive integers:
1  2  2  3  2  4  2  4  3  4  2  6  2  4  4  5  2  6  2  6
4  4  2  8  3  4  4  6  2  8  2  6  4  4  4  9  2  4  4  8
2  8  2  6  6  4  2 10  3  6  4  6  2  8  4  8  4  4  2 12
2  4  6  7  4  8  2  6  4  8  2 12  2  4  6  6  4  8  2 10
5  4  2 12  4  4  4  8  2 12  4  6  4  4  4 12  2  6  6  9```

## C++

`#include <iomanip>#include <iostream> // See https://en.wikipedia.org/wiki/Divisor_functionunsigned int divisor_count(unsigned int n) {    unsigned int total = 1;    // Deal with powers of 2 first    for (; (n & 1) == 0; n >>= 1)        ++total;    // Odd prime factors up to the square root    for (unsigned int p = 3; p * p <= n; p += 2) {        unsigned int count = 1;        for (; n % p == 0; n /= p)            ++count;        total *= count;    }    // If n > 1 then it's prime    if (n > 1)        total *= 2;    return total;} int main() {    const unsigned int limit = 100;    std::cout << "Count of divisors for the first " << limit << " positive integers:\n";    for (unsigned int n = 1; n <= limit; ++n) {        std::cout << std::setw(3) << divisor_count(n);        if (n % 20 == 0)            std::cout << '\n';    }}`
Output:
```Count of divisors for the first 100 positive integers:
1  2  2  3  2  4  2  4  3  4  2  6  2  4  4  5  2  6  2  6
4  4  2  8  3  4  4  6  2  8  2  6  4  4  4  9  2  4  4  8
2  8  2  6  6  4  2 10  3  6  4  6  2  8  4  8  4  4  2 12
2  4  6  7  4  8  2  6  4  8  2 12  2  4  6  6  4  8  2 10
5  4  2 12  4  4  4  8  2 12  4  6  4  4  4 12  2  6  6  9
```

## D

Translation of: C++
`import std.stdio; // See https://en.wikipedia.org/wiki/Divisor_functionuint divisor_count(uint n) {    uint total = 1;    // Deal with powers of 2 first    for (; (n & 1) == 0; n >>= 1) {        ++total;    }    // Odd prime factors up to the square root    for (uint p = 3; p * p <= n; p += 2) {        uint count = 1;        for (; n % p == 0; n /= p) {            ++count;        }        total *= count;    }    // If n > 1 then it's prime    if (n > 1) {        total *= 2;    }    return total;} void main() {    immutable limit = 100;    writeln("Count of divisors for the first ", limit, " positive integers:");    for (uint n = 1; n <= limit; ++n) {        writef("%3d", divisor_count(n));        if (n % 20 == 0) {            writeln;        }    }}`
Output:
```Count of divisors for the first 100 positive integers:
1  2  2  3  2  4  2  4  3  4  2  6  2  4  4  5  2  6  2  6
4  4  2  8  3  4  4  6  2  8  2  6  4  4  4  9  2  4  4  8
2  8  2  6  6  4  2 10  3  6  4  6  2  8  4  8  4  4  2 12
2  4  6  7  4  8  2  6  4  8  2 12  2  4  6  6  4  8  2 10
5  4  2 12  4  4  4  8  2 12  4  6  4  4  4 12  2  6  6  9```

## Delphi

Translation of: Go
` program Tau_function; {\$APPTYPE CONSOLE} uses  System.SysUtils; function CountDivisors(n: Integer): Integer;begin  Result := 0;  var i := 1;  var k := 2;  if (n mod 2) = 0 then    k := 1;   while i * i <= n do  begin    if (n mod i) = 0 then    begin      inc(Result);      var j := n div i;      if j <> i then        inc(Result);    end;    inc(i, k);  end;end; begin  writeln('The tau functions for the first 100 positive integers are:');  for var i := 1 to 100 do  begin    write(CountDivisors(i): 2, ' ');    if (i mod 20) = 0 then      writeln;  end;  readln;end.`

## Dyalect

Translation of: Swift
`func divisorCount(number) {    var n = number    var total = 1     while (n &&& 1) == 0 {        total += 1        n >>>= 1    }     var p = 3    while p * p <= n {        var count = 1        while n % p == 0 {            count += 1            n /= p        }        total *= count        p += 2    }     if n > 1 {        total *= 2    }     total} let limit = 100print("Count of divisors for the first \(limit) positive integers:")for n in 1..limit {    print(divisorCount(number: n).toString().padLeft(2, ' ') + " ", terminator: "")    print() when n % 20 == 0}`
Output:
```Count of divisors for the first 100 positive integers:
1  2  2  3  2  4  2  4  3  4  2  6  2  4  4  5  2  6  2  6
4  4  2  8  3  4  4  6  2  8  2  6  4  4  4  9  2  4  4  8
2  8  2  6  6  4  2 10  3  6  4  6  2  8  4  8  4  4  2 12
2  4  6  7  4  8  2  6  4  8  2 12  2  4  6  6  4  8  2 10
5  4  2 12  4  4  4  8  2 12  4  6  4  4  4 12  2  6  6  9 ```

## F#

This task uses Extensible Prime Generator (F#).

` // Tau function. Nigel Galloway: March 10th., 2021let tau u=let P=primes32()          let rec fN g=match u%g with 0->g |_->fN(Seq.head P)          let rec fG n i g e l=match n=u,u%l with (true,_)->e |(_,0)->fG (n*i) i g (e+g)(l*i) |_->let q=fN(Seq.head P) in fG (n*q) q e (e+e) (q*q)          let n=Seq.head P in fG 1 n 1 1 n[1..100]|>Seq.iter(tau>>printf "%d "); printfn "" `
Output:
```1 2 2 3 2 4 2 4 3 4 2 6 2 4 4 5 2 6 2 6 4 4 2 8 3 4 4 6 2 8 2 6 4 4 4 9 2 4 4 8 2 8 2 6 6 4 2 10 3 6 4 6 2 8 4 8 4 4 2 12 2 4 6 7 4 8 2 6 4 8 2 12 2 4 6 6 4 8 2 10 5 4 2 12 4 4 4 8 2 12 4 6 4 4 4 12 2 6 6 9
```

## Factor

Works with: Factor version 0.99 2020-08-14
`USING: assocs formatting io kernel math math.primes.factorsmath.ranges sequences sequences.extras ; ERROR: nonpositive n ; : (tau) ( n -- count )    group-factors values [ 1 + ] map-product ; inline : tau ( n -- count ) dup 0 > [ (tau) ] [ nonpositive ] if ; "Number of divisors for integers 1-100:" print nl" n   |                   tau(n)" print"-----+-----------------------------------------" print1 100 10 <range> [    [ "%2d   |" printf ]    [ dup 10 + [a,b) [ tau "%4d" printf ] each nl ] bi] each`
Output:
```Number of divisors for integers 1-100:

n   |                   tau(n)
-----+-----------------------------------------
1   |   1   2   2   3   2   4   2   4   3   4
11   |   2   6   2   4   4   5   2   6   2   6
21   |   4   4   2   8   3   4   4   6   2   8
31   |   2   6   4   4   4   9   2   4   4   8
41   |   2   8   2   6   6   4   2  10   3   6
51   |   4   6   2   8   4   8   4   4   2  12
61   |   2   4   6   7   4   8   2   6   4   8
71   |   2  12   2   4   6   6   4   8   2  10
81   |   5   4   2  12   4   4   4   8   2  12
91   |   4   6   4   4   4  12   2   6   6   9
```

## Forth

Translation of: C++
`: divisor_count ( n -- n )  1 >r  begin    dup 2 mod 0=  while    r> 1+ >r    2/  repeat  3  begin    2dup dup * >=  while    1 >r    begin      2dup mod 0=    while      r> 1+ >r      tuck / swap    repeat    2r> * >r    2 +  repeat  drop 1 > if r> 2* else r> then ; : print_divisor_counts ( n -- )  ." Count of divisors for the first " dup . ." positive integers:" cr  1+ 1 do    i divisor_count 2 .r    i 20 mod 0= if cr else space then  loop ; 100 print_divisor_countsbye`
Output:
```Count of divisors for the first 100 positive integers:
1  2  2  3  2  4  2  4  3  4  2  6  2  4  4  5  2  6  2  6
4  4  2  8  3  4  4  6  2  8  2  6  4  4  4  9  2  4  4  8
2  8  2  6  6  4  2 10  3  6  4  6  2  8  4  8  4  4  2 12
2  4  6  7  4  8  2  6  4  8  2 12  2  4  6  6  4  8  2 10
5  4  2 12  4  4  4  8  2 12  4  6  4  4  4 12  2  6  6  9
```

## FreeBASIC

`function numdiv( n as uinteger ) as uinteger    dim as uinteger c = 1    for i as uinteger = 1 to (n+1)\2        if n mod i = 0 then c += 1    next i    if n=1 then c-=1    return cend function for i as uinteger = 1 to 100   print numdiv(i),   if i mod 10 = 0 then printnext i`
Output:
```1             2             2             3             2             4             2             4             3             4
2             6             2             4             4             5             2             6             2             6
4             4             2             8             3             4             4             6             2             8
2             6             4             4             4             9             2             4             4             8
2             8             2             6             6             4             2             10            3             6
4             6             2             8             4             8             4             4             2             12
2             4             6             7             4             8             2             6             4             8
2             12            2             4             6             6             4             8             2             10
5             4             2             12            4             4             4             8             2             12
4             6             4             4             4             12            2             6             6             9```

## Go

`package main import "fmt" func countDivisors(n int) int {    count := 0    i := 1    k := 2    if n%2 == 0 {        k = 1    }    for i*i <= n {        if n%i == 0 {            count++            j := n / i            if j != i {                count++            }        }        i += k    }    return count} func main() {    fmt.Println("The tau functions for the first 100 positive integers are:")    for i := 1; i <= 100; i++ {        fmt.Printf("%2d  ", countDivisors(i))        if i%20 == 0 {            fmt.Println()        }    }}`
Output:
```The tau functions for the first 100 positive integers are:
1   2   2   3   2   4   2   4   3   4   2   6   2   4   4   5   2   6   2   6
4   4   2   8   3   4   4   6   2   8   2   6   4   4   4   9   2   4   4   8
2   8   2   6   6   4   2  10   3   6   4   6   2   8   4   8   4   4   2  12
2   4   6   7   4   8   2   6   4   8   2  12   2   4   6   6   4   8   2  10
5   4   2  12   4   4   4   8   2  12   4   6   4   4   4  12   2   6   6   9
```

## GW-BASIC

`10 FOR N = 1 TO 10020 IF N < 3 THEN T=N: GOTO 7030 T=240 FOR A = 2 TO INT( (N+1)/2 )50 IF N MOD A = 0 THEN T = T + 160 NEXT A70 PRINT T;80 IF N MOD 10 = 0 THEN PRINT90 NEXT N`
Output:
``` 1  2  2  3  2  4  2  4  3  4
2  6  2  4  4  5  2  6  2  6
4  4  2  8  3  4  4  6  2  8
2  6  4  4  4  9  2  4  4  8
2  8  2  6  6  4  2  10  3  6
4  6  2  8  4  8  4  4  2  12
2  4  6  7  4  8  2  6  4  8
2  12  2  4  6  6  4  8  2  10
5  4  2  12  4  4  4  8  2  12
4  6  4  4  4  12  2  6  6  9
```

`tau :: Integral a => a -> atau n | n <= 0 = error "Not a positive integer"tau n = go 0 (1, 1)    where    yo i = (i, i * i)    go r (i, ii)        | n < ii = r        | n == ii = r + 1        | 0 == mod n i = go (r + 2) (yo \$ i + 1)        | otherwise = go r (yo \$ i + 1) main = print \$ map tau [1..100]`
Output:
`[1,2,2,3,2,4,2,4,3,4,2,6,2,4,4,5,2,6,2,6,4,4,2,8,3,4,4,6,2,8,2,6,4,4,4,9,2,4,4,8,2,8,2,6,6,4,2,10,3,6,4,6,2,8,4,8,4,4,2,12,2,4,6,7,4,8,2,6,4,8,2,12,2,4,6,6,4,8,2,10,5,4,2,12,4,4,4,8,2,12,4,6,4,4,4,12,2,6,6,9]`

## Java

Translation of: D
`public class TauFunction {    private static long divisorCount(long n) {        long total = 1;        // Deal with powers of 2 first        for (; (n & 1) == 0; n >>= 1) {            ++total;        }        // Odd prime factors up to the square root        for (long p = 3; p * p <= n; p += 2) {            long count = 1;            for (; n % p == 0; n /= p) {                ++count;            }            total *= count;        }        // If n > 1 then it's prime        if (n > 1) {            total *= 2;        }        return total;    }     public static void main(String[] args) {        final int limit = 100;        System.out.printf("Count of divisors for the first %d positive integers:\n", limit);        for (long n = 1; n <= limit; ++n) {            System.out.printf("%3d", divisorCount(n));            if (n % 20 == 0) {                System.out.println();            }        }    }}`
Output:
```Count of divisors for the first 100 positive integers:
1  2  2  3  2  4  2  4  3  4  2  6  2  4  4  5  2  6  2  6
4  4  2  8  3  4  4  6  2  8  2  6  4  4  4  9  2  4  4  8
2  8  2  6  6  4  2 10  3  6  4  6  2  8  4  8  4  4  2 12
2  4  6  7  4  8  2  6  4  8  2 12  2  4  6  6  4  8  2 10
5  4  2 12  4  4  4  8  2 12  4  6  4  4  4 12  2  6  6  9```

## jq

Works with: jq

Works with gojq, the Go implementation of jq

Preliminaries

See https://rosettacode.org/wiki/Sum_of_divisors#jq for the definition of `divisors` used here.
`def count(s): reduce s as \$x (0; .+1); # For pretty-printingdef 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] + .;`

`[range(1;101) | count(divisors)]| nwise(10) | map(lpad(4)) | join("")`
Output:
```   1   2   2   3   2   4   2   4   3   4
2   6   2   4   4   5   2   6   2   6
4   4   2   8   3   4   4   6   2   8
2   6   4   4   4   9   2   4   4   8
2   8   2   6   6   4   2  10   3   6
4   6   2   8   4   8   4   4   2  12
2   4   6   7   4   8   2   6   4   8
2  12   2   4   6   6   4   8   2  10
5   4   2  12   4   4   4   8   2  12
4   6   4   4   4  12   2   6   6   9
```

## Julia

`using Primes function numfactors(n)    f = [one(n)]    for (p, e) in factor(n)        f = reduce(vcat, [f * p^j for j in 1:e], init = f)    end    length(f)end for i in 1:100    print(rpad(numfactors(i), 3), i % 25 == 0 ? " \n" : " ")end `
Output:
```1   2   2   3   2   4   2   4   3   4   2   6   2   4   4   5   2   6   2   6   4   4   2   8   3
4   4   6   2   8   2   6   4   4   4   9   2   4   4   8   2   8   2   6   6   4   2   10  3   6
4   6   2   8   4   8   4   4   2   12  2   4   6   7   4   8   2   6   4   8   2   12  2   4   6
6   4   8   2   10  5   4   2   12  4   4   4   8   2   12  4   6   4   4   4   12  2   6   6   9
```

## Lua

Translation of: Java
`function divisorCount(n)    local total = 1    -- Deal with powers of 2 first    while (n & 1) == 0 do        total = total + 1        n = math.floor(n / 2)    end    -- Odd prime factors up tot eh square root    local p = 3    while p * p <= n do        local count = 1        while n % p == 0 do            count = count + 1            n = n / p        end        total = total * count        p = p + 2    end    -- If n > 1 then it's prime    if n > 1 then        total = total * 2    end    return totalend limit = 100print("Count of divisors for the first " .. limit .. " positive integers:")for n=1,limit do    io.write(string.format("%3d", divisorCount(n)))    if n % 20 == 0 then        print()    endend`
Output:
```Count of divisors for the first 100 positive integers:
1  2  2  3  2  4  2  4  3  4  2  6  2  4  4  5  2  6  2  6
4  4  2  8  3  4  4  6  2  8  2  6  4  4  4  9  2  4  4  8
2  8  2  6  6  4  2 10  3  6  4  6  2  8  4  8  4  4  2 12
2  4  6  7  4  8  2  6  4  8  2 12  2  4  6  6  4  8  2 10
5  4  2 12  4  4  4  8  2 12  4  6  4  4  4 12  2  6  6  9```

## Mathematica/Wolfram Language

`DivisorSum[#, 1 &] & /@ Range[100]`
Output:
`{1,2,2,3,2,4,2,4,3,4,2,6,2,4,4,5,2,6,2,6,4,4,2,8,3,4,4,6,2,8,2,6,4,4,4,9,2,4,4,8,2,8,2,6,6,4,2,10,3,6,4,6,2,8,4,8,4,4,2,12,2,4,6,7,4,8,2,6,4,8,2,12,2,4,6,6,4,8,2,10,5,4,2,12,4,4,4,8,2,12,4,6,4,4,4,12,2,6,6,9}`

## Nim

`import math, strutils func divcount(n: Natural): Natural =  for i in 1..sqrt(n.toFloat).int:    if n mod i == 0:      inc result      if n div i != i: inc result echo "Count of divisors for the first 100 positive integers:"for i in 1..100:  stdout.write (\$divcount(i)).align(3)  if i mod 20 == 0: echo()`
Output:
```Count of divisors for the first 100 positive integers:
1  2  2  3  2  4  2  4  3  4  2  6  2  4  4  5  2  6  2  6
4  4  2  8  3  4  4  6  2  8  2  6  4  4  4  9  2  4  4  8
2  8  2  6  6  4  2 10  3  6  4  6  2  8  4  8  4  4  2 12
2  4  6  7  4  8  2  6  4  8  2 12  2  4  6  6  4  8  2 10
5  4  2 12  4  4  4  8  2 12  4  6  4  4  4 12  2  6  6  9```

## PARI/GP

`vector(100,X,numdiv(X))`
Output:
```
[1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8, 2, 6, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 10, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 12, 2, 4, 6, 7, 4, 8, 2, 6, 4, 8, 2, 12, 2, 4, 6, 6, 4, 8, 2, 10, 5, 4, 2, 12, 4, 4, 4, 8, 2, 12, 4, 6, 4, 4, 4, 12, 2, 6, 6, 9]```

## Perl

Library: ntheory
`use strict;use warnings;use feature 'say';use ntheory 'divisors'; my @x;push @x, scalar divisors(\$_) for 1..100; say "Tau function - first 100:\n" .    ((sprintf "@{['%4d' x 100]}", @x[0..100-1]) =~ s/(.{80})/\$1\n/gr);`
Output:
```   1   2   2   3   2   4   2   4   3   4   2   6   2   4   4   5   2   6   2   6
4   4   2   8   3   4   4   6   2   8   2   6   4   4   4   9   2   4   4   8
2   8   2   6   6   4   2  10   3   6   4   6   2   8   4   8   4   4   2  12
2   4   6   7   4   8   2   6   4   8   2  12   2   4   6   6   4   8   2  10
5   4   2  12   4   4   4   8   2  12   4   6   4   4   4  12   2   6   6   9```

## Phix

### imperative

```for i=1 to 100 do
printf(1,"%3d",{length(factors(i,1))})
if remainder(i,20)=0 then puts(1,"\n") end if
end for
```
Output:
```  1  2  2  3  2  4  2  4  3  4  2  6  2  4  4  5  2  6  2  6
4  4  2  8  3  4  4  6  2  8  2  6  4  4  4  9  2  4  4  8
2  8  2  6  6  4  2 10  3  6  4  6  2  8  4  8  4  4  2 12
2  4  6  7  4  8  2  6  4  8  2 12  2  4  6  6  4  8  2 10
5  4  2 12  4  4  4  8  2 12  4  6  4  4  4 12  2  6  6  9
```

### functional

same output

```sequence r = apply(apply(true,factors,{tagset(100),{1}}),length)
puts(1,join_by(apply(true,sprintf,{{"%3d"},r}),1,20,""))
```

## PureBasic

`If OpenConsole()  For i=1 To 100    If i<3 : Print(RSet(Str(i),4)) : Continue :EndIf    c=2    For j=2 To i/2+1 : c+Bool(i%j=0) : Next    Print(RSet(Str(c),4))    If i%10=0 : PrintN("") : EndIf  Next  Input()EndIfEnd`
Output:
```   1   2   2   3   2   4   2   4   3   4
2   6   2   4   4   5   2   6   2   6
4   4   2   8   3   4   4   6   2   8
2   6   4   4   4   9   2   4   4   8
2   8   2   6   6   4   2  10   3   6
4   6   2   8   4   8   4   4   2  12
2   4   6   7   4   8   2   6   4   8
2  12   2   4   6   6   4   8   2  10
5   4   2  12   4   4   4   8   2  12
4   6   4   4   4  12   2   6   6   9```

## Python

### Using prime factorization

`def factorize(n):    assert(isinstance(n, int))    if n < 0:         n = -n     if n < 2:         return     k = 0     while 0 == n%2:         k += 1         n //= 2     if 0 < k:         yield (2,k)     p = 3     while p*p <= n:         k = 0         while 0 == n%p:             k += 1             n //= p         if 0 < k:             yield (p,k)        p += 2     if 1 < n:         yield (n,1)  def tau(n):     assert(n != 0)     ans = 1     for (p,k) in factorize(n):         ans *= 1 + k    return ans if __name__ == "__main__":    print([tau(n) for n in range(1,101)])`

### Finding divisors efficiently

`def tau(n):    assert(isinstance(n, int) and 0 < n)    ans, i, j = 0, 1, 1    while i*i <= n:        if 0 == n%i:            ans += 1            j = n//i            if j != i:                ans += 1        i += 1    return ans if __name__ == "__main__":    print([tau(n) for n in range(1,101)])`
Output:
`[1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8, 2, 6, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 10, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 12, 2, 4, 6, 7, 4, 8, 2, 6, 4, 8, 2, 12, 2, 4, 6, 6, 4, 8, 2, 10, 5, 4, 2, 12, 4, 4, 4, 8, 2, 12, 4, 6, 4, 4, 4, 12, 2, 6, 6, 9]`

### Choosing the right abstraction

Yet another exercise in defining a divisors function.

`'''The number of divisors of n''' from itertools import count, islicefrom math import floor, sqrt  # oeisA000005 :: [Int]def oeisA000005():    '''tau(n) (also called d(n) or sigma_0(n)),       the number of divisors of n.    '''    return map(tau, count(1))  # tau :: Int -> Intdef tau(n):    '''The number of divisors of n.    '''    return len(divisors(n))  # ------------------------- TEST -------------------------# main :: IO ()def main():    '''The first 100 terms of OEIS A000005.       (Shown in rows of 10)    '''    terms = take(100)(        oeisA000005()    )    columnWidth = 1 + len(str(max(terms)))     print(        '\n'.join(            ''.join(                str(term).rjust(columnWidth)                for term in row            )            for row in chunksOf(10)(terms)        )    )  # ----------------------- GENERIC ------------------------ # chunksOf :: Int -> [a] -> [[a]]def chunksOf(n):    '''A series of lists of length n, subdividing the       contents of xs. Where the length of xs is not evenly       divible, the final list will be shorter than n.    '''    def go(xs):        return [            xs[i:n + i] for i in range(0, len(xs), n)        ] if 0 < n else None    return go  # divisors :: Int -> [Int]def divisors(n):    '''List of all divisors of n including n itself.    '''    root = floor(sqrt(n))    lows = [x for x in range(1, 1 + root) if 0 == n % x]    return lows + [n // x for x in reversed(lows)][        (1 if n == (root * root) else 0):    ]  # take :: Int -> [a] -> [a]# take :: Int -> String -> Stringdef take(n):    '''The prefix of xs of length n,       or xs itself if n > length xs.    '''    def go(xs):        return (            xs[0:n]            if isinstance(xs, (list, tuple))            else list(islice(xs, n))        )    return go  # MAIN ---if __name__ == '__main__':    main()`
Output:
```  1  2  2  3  2  4  2  4  3  4
2  6  2  4  4  5  2  6  2  6
4  4  2  8  3  4  4  6  2  8
2  6  4  4  4  9  2  4  4  8
2  8  2  6  6  4  2 10  3  6
4  6  2  8  4  8  4  4  2 12
2  4  6  7  4  8  2  6  4  8
2 12  2  4  6  6  4  8  2 10
5  4  2 12  4  4  4  8  2 12
4  6  4  4  4 12  2  6  6  9```

## Quackery

`factors` is defined at Factors of an integer#Quackery.

`  [ factors size ] is tau ( n --> n )   [] []   100 times [ i^ 1+ tau join ]  witheach [ number\$ nested join ]  70 wrap\$  `
Output:
```1 2 2 3 2 4 2 4 3 4 2 6 2 4 4 5 2 6 2 6 4 4 2 8 3 4 4 6 2 8 2 6 4 4 4
9 2 4 4 8 2 8 2 6 6 4 2 10 3 6 4 6 2 8 4 8 4 4 2 12 2 4 6 7 4 8 2 6 4
8 2 12 2 4 6 6 4 8 2 10 5 4 2 12 4 4 4 8 2 12 4 6 4 4 4 12 2 6 6 9
```

## Raku

Yet more tasks that are tiny variations of each other. Tau function, Tau number, Sum of divisors and Product of divisors all use code with minimal changes. What the heck, post 'em all.

`use Prime::Factor:ver<0.3.0+>;use Lingua::EN::Numbers; say "\nTau function - first 100:\n",        # ID(1..*).map({ +.&divisors })[^100]\          # the task.batch(20)».fmt("%3d").join("\n");          # display formatting say "\nTau numbers - first 100:\n",         # ID(1..*).grep({ \$_ %% +.&divisors })[^100]\   # the task.batch(10)».&comma».fmt("%5s").join("\n");  # display formatting say "\nDivisor sums - first 100:\n",        # ID(1..*).map({ [+] .&divisors })[^100]\       # the task.batch(20)».fmt("%4d").join("\n");          # display formatting say "\nDivisor products - first 100:\n",    # ID(1..*).map({ [×] .&divisors })[^100]\       # the task.batch(5)».&comma».fmt("%16s").join("\n");  # display formatting`
Output:
```Tau function - first 100:
1   2   2   3   2   4   2   4   3   4   2   6   2   4   4   5   2   6   2   6
4   4   2   8   3   4   4   6   2   8   2   6   4   4   4   9   2   4   4   8
2   8   2   6   6   4   2  10   3   6   4   6   2   8   4   8   4   4   2  12
2   4   6   7   4   8   2   6   4   8   2  12   2   4   6   6   4   8   2  10
5   4   2  12   4   4   4   8   2  12   4   6   4   4   4  12   2   6   6   9

Tau numbers - first 100:
1     2     8     9    12    18    24    36    40    56
60    72    80    84    88    96   104   108   128   132
136   152   156   180   184   204   225   228   232   240
248   252   276   288   296   328   344   348   360   372
376   384   396   424   441   444   448   450   468   472
480   488   492   504   516   536   560   564   568   584
600   612   625   632   636   640   664   672   684   708
712   720   732   776   792   804   808   824   828   852
856   864   872   876   880   882   896   904   936   948
972   996 1,016 1,040 1,044 1,048 1,056 1,068 1,089 1,096

Divisor sums - first 100:
1    3    4    7    6   12    8   15   13   18   12   28   14   24   24   31   18   39   20   42
32   36   24   60   31   42   40   56   30   72   32   63   48   54   48   91   38   60   56   90
42   96   44   84   78   72   48  124   57   93   72   98   54  120   72  120   80   90   60  168
62   96  104  127   84  144   68  126   96  144   72  195   74  114  124  140   96  168   80  186
121  126   84  224  108  132  120  180   90  234  112  168  128  144  120  252   98  171  156  217

Divisor products - first 100:
1                2                3                8                5
36                7               64               27              100
11            1,728               13              196              225
1,024               17            5,832               19            8,000
441              484               23          331,776              125
676              729           21,952               29          810,000
31           32,768            1,089            1,156            1,225
10,077,696               37            1,444            1,521        2,560,000
41        3,111,696               43           85,184           91,125
2,116               47      254,803,968              343          125,000
2,601          140,608               53        8,503,056            3,025
9,834,496            3,249            3,364               59   46,656,000,000
61            3,844          250,047        2,097,152            4,225
18,974,736               67          314,432            4,761       24,010,000
71  139,314,069,504               73            5,476          421,875
438,976            5,929       37,015,056               79    3,276,800,000
59,049            6,724               83  351,298,031,616            7,225
7,396            7,569       59,969,536               89  531,441,000,000
8,281          778,688            8,649            8,836            9,025
782,757,789,696               97          941,192          970,299    1,000,000,000```

## REXX

`/*REXX program counts the number of divisors (tau,  or sigma_0)  up to and including  N.*/parse arg LO HI cols .                           /*obtain optional argument from the CL.*/if   LO=='' |   LO==","  then  LO=   1           /*Not specified?  Then use the default.*/if   HI=='' |   HI==","  then  HI=  LO + 100 - 1 /*Not specified?  Then use the default.*/if cols=='' | cols==","  then cols= 20           /* "      "         "   "   "     "    */w= 2 + (HI>45359)                                /*W:  used to align the output columns.*/say 'The number of divisors  (tau)  for integers up to '    n    " (inclusive):";      saysay '─index─'   center(" tau (number of divisors) ",  cols * (w+1)  +  1,  '─')\$=;                                   c= 0       /*\$:  the output list,  shown ROW/line.*/                do j=LO  to HI;       c= c + 1   /*list # proper divisors (tau) 1 ──► N */                \$= \$  right( tau(j), w)          /*add a tau number to the output list. */                if c//cols \== 0  then iterate   /*Not a multiple of ROW? Don't display.*/                idx= j - cols + 1                /*calculate index value (for this row).*/                say center(idx, 7)    \$;  \$=     /*display partial list to the terminal.*/                end   /*j*/ if \$\==''  then say center(idx+cols, 7)    \$     /*there any residuals left to display ?*/exit 0                                           /*stick a fork in it,  we're all done. *//*──────────────────────────────────────────────────────────────────────────────────────*/tau: procedure; parse arg x 1 y                  /*X  and  \$  are both set from the arg.*/     if x<6  then return 2 + (x==4) - (x==1)     /*some low #s should be handled special*/     odd= x // 2                                 /*check if  X  is odd (remainder of 1).*/     if odd  then       #= 2                     /*Odd?    Assume divisor count of  2.  */             else do;   #= 4;   y= x % 2;   end  /*Even?      "      "      "    "  4.  */                                                 /* [↑]  start with known number of divs*/        do j=3  for x%2-3  by 1+odd  while j<y   /*for odd number,  skip even numbers.  */        if x//j==0  then do                      /*if no remainder, then found a divisor*/                         #= # + 2;   y= x % j    /*bump # of divisors;  calculate limit.*/                         if j>=y  then do;   #= # - 1;   leave;   end   /*reached limit?*/                         end                     /*                     ___             */                    else if j*j>x  then leave    /*only divide up to   √ x              */        end   /*j*/;               return #      /* [↑]  this form of DO loop is faster.*/`
output   when using the default input:
```The number of divisors  (tau)  for integers up to  100  (inclusive):

─index─ ──────────────────────────── tau (number of divisors) ────────────────────────────
1       1   2   2   3   2   4   2   4   3   4   2   6   2   4   4   5   2   6   2   6
21       4   4   2   8   3   4   4   6   2   8   2   6   4   4   4   9   2   4   4   8
41       2   8   2   6   6   4   2  10   3   6   4   6   2   8   4   8   4   4   2  12
61       2   4   6   7   4   8   2   6   4   8   2  12   2   4   6   6   4   8   2  10
81       5   4   2  12   4   4   4   8   2  12   4   6   4   4   4  12   2   6   6   9
```

## Ring

` see "The tau functions for the first 100 positive integers are:" + nl  n = 0num = 0limit = 100while num < limit      n = n + 1      tau = 0      for m = 1 to n          if n%m = 0             tau = tau + 1          ok      next      num = num + 1      if num%10 = 1         see nl      ok      tau = string(tau)      if len(tau) = 1         tau = " " + tau       ok      see "" + tau + " "end `

Output:

```The tau functions for the first 100 positive integers are:

1  2  2  3  2  4  2  4  3  4
2  6  2  4  4  5  2  6  2  6
4  4  2  8  3  4  4  6  2  8
2  6  4  4  4  9  2  4  4  8
2  8  2  6  6  4  2 10  3  6
4  6  2  8  4  8  4  4  2 12
2  4  6  7  4  8  2  6  4  8
2 12  2  4  6  6  4  8  2 10
5  4  2 12  4  4  4  8  2 12
4  6  4  4  4 12  2  6  6  9
```

## Ruby

`require 'prime' def tau(n) = n.prime_division.inject(1){|res, (d, exp)| res *= exp + 1} (1..100).map{|n| tau(n).to_s.rjust(3) }.each_slice(20){|ar| puts ar.join} `
Output:
```  1  2  2  3  2  4  2  4  3  4  2  6  2  4  4  5  2  6  2  6
4  4  2  8  3  4  4  6  2  8  2  6  4  4  4  9  2  4  4  8
2  8  2  6  6  4  2 10  3  6  4  6  2  8  4  8  4  4  2 12
2  4  6  7  4  8  2  6  4  8  2 12  2  4  6  6  4  8  2 10
5  4  2 12  4  4  4  8  2 12  4  6  4  4  4 12  2  6  6  9
```

## Rust

`// returns the highest power of i that is a factor of n,// and n divided by that power of ifn factor_exponent(n: i32, i: i32) -> (i32, i32) {	if n % i == 0 {		let (a, b) = factor_exponent(n / i, i);		(a + 1, b)	} else {		(0, n)	}} fn tau(n: i32) -> i32 {	for i in 2..(n+1) {		if n % i == 0 {			let (count, next) = factor_exponent(n, i);			return (count + 1) * tau(next);		}	}	return 1;} fn main() {	for i in 1..101 {		print!("{} ", tau(i));	}}`

Output:

```1 2 2 3 2 4 2 4 3 4 2 6 2 4 4 5 2 6 2 6 4 4 2 8 3 4 4 6 2 8 2 6 4 4 4 9 2 4 4 8 2 8 2 6 6 4 2 10 3 6 4 6 2 8 4 8 4 4 2 12 2 4 6 7 4 8 2 6 4 8 2 12 2 4 6 6 4 8 2 10 5 4 2 12 4 4 4 8 2 12 4 6 4 4 4 12 2 6 6 9
```

## Sidef

Built-in:

`say { .sigma0 }.map(1..100).join(' ')`
Output:
```1 2 2 3 2 4 2 4 3 4 2 6 2 4 4 5 2 6 2 6 4 4 2 8 3 4 4 6 2 8 2 6 4 4 4 9 2 4 4 8 2 8 2 6 6 4 2 10 3 6 4 6 2 8 4 8 4 4 2 12 2 4 6 7 4 8 2 6 4 8 2 12 2 4 6 6 4 8 2 10 5 4 2 12 4 4 4 8 2 12 4 6 4 4 4 12 2 6 6 9
```

## Swift

`import Foundation // See https://en.wikipedia.org/wiki/Divisor_functionfunc divisorCount(number: Int) -> Int {    var n = number    var total = 1    // Deal with powers of 2 first    while (n & 1) == 0 {        total += 1        n >>= 1    }    // Odd prime factors up to the square root    var p = 3    while p * p <= n {        var count = 1        while n % p == 0 {            count += 1            n /= p        }        total *= count        p += 2    }    // If n > 1 then it's prime    if n > 1 {        total *= 2    }    return total} let limit = 100print("Count of divisors for the first \(limit) positive integers:")for n in 1...limit {    print(String(format: "%3d", divisorCount(number: n)), terminator: "")    if n % 20 == 0 {        print()    }}`
Output:
```Count of divisors for the first 100 positive integers:
1  2  2  3  2  4  2  4  3  4  2  6  2  4  4  5  2  6  2  6
4  4  2  8  3  4  4  6  2  8  2  6  4  4  4  9  2  4  4  8
2  8  2  6  6  4  2 10  3  6  4  6  2  8  4  8  4  4  2 12
2  4  6  7  4  8  2  6  4  8  2 12  2  4  6  6  4  8  2 10
5  4  2 12  4  4  4  8  2 12  4  6  4  4  4 12  2  6  6  9
```

## Tiny BASIC

`    LET N = 0 10 LET N = N + 1    IF N < 3 THEN GOTO 100    LET T = 2    LET A = 1 20 LET A = A + 1    IF (N/A)*A = N THEN LET T = T + 1    IF A<(N+1)/2 THEN GOTO 20 30 PRINT "Tau(",N,") = ",T    IF N<100 THEN GOTO 10    END100 LET T = N    GOTO 30`

## Wren

Library: Wren-math
Library: Wren-fmt
`import "/math" for Intimport "/fmt" for Fmt System.print("The tau functions for the first 100 positive integers are:")for (i in 1..100) {    Fmt.write("\$2d  ", Int.divisors(i).count)    if (i % 20 == 0) System.print()}`
Output:
```The tau functions for the first 100 positive integers are:
1   2   2   3   2   4   2   4   3   4   2   6   2   4   4   5   2   6   2   6
4   4   2   8   3   4   4   6   2   8   2   6   4   4   4   9   2   4   4   8
2   8   2   6   6   4   2  10   3   6   4   6   2   8   4   8   4   4   2  12
2   4   6   7   4   8   2   6   4   8   2  12   2   4   6   6   4   8   2  10
5   4   2  12   4   4   4   8   2  12   4   6   4   4   4  12   2   6   6   9
```