I'm working on modernizing Rosetta Code's infrastructure. Starting with communications. Please accept this time-limited open invite to RC's Slack.. --Michael Mol (talk) 20:59, 30 May 2020 (UTC)

# Tau number

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

A Tau number is a positive integer divisible by the count of its positive divisors.

Show the first   100   Tau numbers. The numbers shall be generated during run-time (i.e. the code may not contain string literals, sets/arrays of integers, or alike).

## 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 F is_tau_number(n)   I n <= 0      R 0B   R 0 == n % tau(n) V n = 1[Int] ansL ans.len < 100   I is_tau_number(n)      ans.append(n)   n++print(ans)`
Output:
```[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, 1016, 1040, 1044, 1048, 1056, 1068, 1089, 1096]
```

## Action!

`CARD FUNC DivisorCount(CARD n)  CARD result,p,count   result=1  WHILE (n&1)=0  DO    result==+1    n=n RSH 1  OD   p=3  WHILE p*p<=n  DO    count=1    WHILE n MOD p=0    DO      count==+1      n==/p    OD    result==*count    p==+2  OD   IF n>1 THEN    result==*2  FIRETURN (result) PROC Main()  CARD n=[1],max=[100],count=[0],divCount   WHILE count<max  DO    divCount=DivisorCount(n)    IF n MOD divCount=0 THEN      PrintC(n) Put(32)      count==+1    FI    n==+1  ODRETURN`
Output:
```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 1016 1040 1044 1048 1056 1068 1089 1096
```

## ALGOL 68

Translation of: C++
`BEGIN # find tau numbers - numbers divisible by the count of theoir divisors #    # calculates the number of divisors of v                                 #    PROC divisor count = ( INT v )INT:         BEGIN            INT total := 1, n := v;            # Deal with powers of 2 first #            WHILE NOT ODD n DO                total +:= 1;                n  OVERAB 2            OD;            # Odd prime factors up to the square root #            INT p := 1;            WHILE  p +:= 2;                   ( p * p ) <= n            DO                INT count := 1;                WHILE n MOD p = 0 DO                    count +:= 1;                    n  OVERAB p                OD;                total *:= count            OD;            # If n > 1 then it's prime #            IF n > 1 THEN total *:= 2 FI;            total         END # divisor count #;    BEGIN        INT tau limit  = 100;        INT tau count := 0;        print( ( "The first ", whole( tau limit, 0 ), " tau numbers:", newline ) );        FOR n WHILE tau count < tau limit DO            IF n MOD divisor count( n ) = 0 THEN                tau count +:= 1;                print( ( whole( n, -6 ) ) );                IF tau count MOD 10 = 0 THEN print( ( newline ) ) FI            FI        OD    ENDEND`
Output:
```The first 100 tau numbers:
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  1016  1040  1044  1048  1056  1068  1089  1096
```

## ALGOL-M

`begin integer array dcount[1:1100];integer i, j, n; integer function mod(a,b);integer a,b;mod := a-a/b*b; % Calculate counts of divisors for 1 .. 1100 %for i := 1 step 1 until 1100 do dcount[i] := 1;for i := 2 step 1 until 1100 dobegin    j := i;    while j <= 1100 do    begin        dcount[j] := dcount[j] + 1;        j := j + i;    end;end; n := 0;i := 1;while n < 100 dobegin    if mod(i, dcount[i])=0 then    begin        if mod(n, 10)=0            then write(i)            else writeon(i);        n := n + 1;    end;    i := i + 1;end;end`
Output:
```     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  1016  1040  1044  1048  1056  1068  1089  1096```

## APL

Works with: Dyalog APL
`(⊢(/⍨)(0=(0+.=⍳|⊢)|⊢)¨)⍳ 1096`
Output:
```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 1016 1040 1044 1048 1056 1068 1089 1096```

## 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, counter, astidset output to {"First 100 tau numbers:"}set n to 0set counter to 0repeat until (counter = 100)    set n to n + 1    if (n mod (factorCount(n)) = 0) then        set counter to counter + 1        if (counter mod 20 = 1) then set end of output to linefeed        set end of output to text -5 thru -1 of ("    " & n)    end ifend 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:
`"First 100 tau numbers:    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 1016 1040 1044 1048 1056 1068 1089 1096"`

## Arturo

`tau: function [x] -> size factors x found: 0i:1while [found<100][    if 0 = i % tau i [        prints pad to :string i 5        found: found + 1        if 0 = found % 10 -> print ""    ]    i: i + 1]`
Output:
```    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 1016 1040 1044 1048 1056 1068 1089 1096```

## AWK

` # syntax: GAWK -f TAU_NUMBER.AWKBEGIN {    print("The first 100 tau numbers:")    while (count < 100) {      i++      if (i % count_divisors(i) == 0) {        printf("%4d ",i)        if (++count % 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 first 100 tau numbers:
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 1016 1040 1044 1048 1056 1068 1089 1096
```

## BASIC

`10 DEFINT A-Z20 S=0: N=130 C=140 IF N<>1 THEN FOR I=1 TO N/2: C=C-(N MOD I=0): NEXT50 IF N MOD C=0 THEN PRINT N,: S=S+160 N=N+170 IF S<100 THEN 3080 END`
Output:
``` 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           1016          1040          1044
1048          1056          1068          1089          1096```

### BASIC256

`print "The first 100 tau numbers are:" n = 0num = 0limit = 100while num < limit	n += 1	tau = 0	for m = 1 to n		if n mod m = 0 then tau += 1	next m	if n mod tau = 0 then		num += 1		if num mod 10 = 1 then print		print n; "  ";	end ifend whileend`

### QBasic

Works with: QBasic version 1.1
`PRINT "The first 100 tau numbers are:" n = 0num = 0limit = 100DO      n = n + 1      tau = 0      FOR m = 1 TO n          IF n MOD m = 0 THEN tau = tau + 1      NEXT m      IF n MOD tau = 0 THEN         num = num + 1         IF num MOD 10 = 1 THEN PRINT         PRINT USING " ####"; n; '""; n; " ";      END IFLOOP WHILE num < limitEND`

### True BASIC

`LET n = 0LET num = 0LET limit = 100DO   LET n = n + 1   LET tau = 0   FOR m = 1 TO n       IF REMAINDER(n, m) = 0 THEN LET tau = tau + 1   NEXT m   IF REMAINDER(n, tau) = 0 THEN      LET num = num + 1      IF REMAINDER(num, 10) = 1 THEN PRINT      PRINT ""; n; " ";   END IFLOOP WHILE num < limitEND`

### Yabasic

`print "The first 100 tau numbers are:" n = 0num = 0limit = 100while num < limit      n = n + 1      tau = 0      for m = 1 to n          if mod(n, m) = 0 then tau = tau + 1 : fi      next m      if mod(n, tau) = 0 then         num = num + 1         if mod(num, 10) = 1 then print : fi         print n using "####";      end ifwendprintend`

## BCPL

`get "libhdr" // Count the divisors of 1..Nlet divcounts(v, n) be\$(  // Every positive number is divisible by 1    for i=1 to n do v!i := 1;    for i=2 to n do     \$(  let j = i        while j <= n do        \$(  // J is divisible by I            v!j := v!j + 1            j := j + i        \$)    \$)\$) // Given a stored vector of divisors counts, is a number a tau number?let tau(v, i) = i rem v!i = 0 let start() be\$(  let dvec = vec 1100    let n, seen = 1, 0     divcounts(dvec, 1100) // find amount of divisors for each number    while seen < 100 do    \$(  if tau(dvec, n) then        \$(  writed(n, 5)            seen := seen + 1            if seen rem 10 = 0 then wrch('*N')        \$)        n := n + 1    \$)\$)`
Output:
```    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 1016 1040 1044 1048 1056 1068 1089 1096```

## C

Translation of: C++
`#include <stdio.h> unsigned int divisor_count(unsigned int n) {    unsigned int total = 1;    unsigned int p;     // Deal with powers of 2 first    for (; (n & 1) == 0; n >>= 1) {        ++total;    }    // Odd prime factors up to the square root    for (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 count = 0;    unsigned int n;     printf("The first %d tau numbers are:\n", limit);    for (n = 1; count < limit; ++n) {        if (n % divisor_count(n) == 0) {            printf("%6d", n);            ++count;            if (count % 10 == 0) {                printf("\n");            }        }    }     return 0;}`
Output:
```The first 100 tau numbers are:
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  1016  1040  1044  1048  1056  1068  1089  1096```

## 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 << "The first " << limit << " tau numbers are:\n";    unsigned int count = 0;    for (unsigned int n = 1; count < limit; ++n) {        if (n % divisor_count(n) == 0) {            std::cout << std::setw(6) << n;            ++count;            if (count % 10 == 0)                std::cout << '\n';        }    }}`
Output:
```The first 100 tau numbers are:
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  1016  1040  1044  1048  1056  1068  1089  1096
```

## CLU

`% Count the divisors of [1..N]count_divisors = proc (n: int) returns (sequence[int])    divs: array[int] := array[int]\$fill(1, n, 1)    for i: int in int\$from_to(2, n) do        for j: int in int\$from_to_by(i, n, i) do            divs[j] := divs[j] + 1        end    end    return(sequence[int]\$a2s(divs))end count_divisors % Find Tau numbers up to a given limittau_numbers = iter (lim: int) yields (int)    divs: sequence[int] := count_divisors(lim)    n: int := 0    while n < lim do        n := n + 1        if n // divs[n] = 0 then yield(n) end    endend tau_numbers % Show the first 100 Tau numbersstart_up = proc ()    po: stream := stream\$primary_output()    seen: int := 0     for n: int in tau_numbers(1100) do        seen := seen + 1        stream\$putright(po, int\$unparse(n), 5)        if seen // 10 = 0 then stream\$putl(po, "") end        if seen >= 100 then break end    endend start_up`
Output:
```    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 1016 1040 1044 1048 1056 1068 1089 1096```

## Cowgol

`include "cowgol.coh"; # Get count of positive divisors of numbersub pos_div(num: uint16): (count: uint16) is    count := 1;    if num != 1 then        var cur: uint16 := 1;        while cur <= num/2 loop            if num % cur == 0 then                count := count + 1;            end if;            cur := cur + 1;        end loop;    end if;end sub; # Print first 100 Tau numbersvar nums: uint8 := 0;var cur: uint16 := 0;var col: uint16 := 10;while nums < 100 loop    cur := cur + 1;    if cur % pos_div(cur) == 0 then        print_i16(cur);        col := col - 1;        if col == 0 then            print_nl();            col := 10;        else            print_char('\t');        end if;        nums := nums + 1;    end if;end loop;`
Output:
```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     1016    1040    1044    1048    1056    1068    1089    1096```

## D

Translation of: C++
`import std.stdio; uint 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("The first ", limit, " tau numbers are:");    uint count = 0;    for (uint n = 1; count < limit; ++n) {        if (n % divisor_count(n) == 0) {            writef("%6d", n);            ++count;            if (count % 10 == 0) {                writeln;            }        }    }}`
Output:
```The first 100 tau numbers are:
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  1016  1040  1044  1048  1056  1068  1089  1096```

## Delphi

Translation of: Go
` program Tau_number; {\$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 first 100 tau numbers are:');  var count := 0;  var i := 1;  while count < 100 do  begin    var tf := CountDivisors(i);    if i mod tf = 0 then    begin      write(format('%4d ', [i]));      inc(count);      if count mod 10 = 0 then        writeln;    end;    inc(i);  end;   {\$IFNDEF UNIX}  readln; {\$ENDIF}end.`

## Draco

`/* Generate a table of the amount of divisors for each number */proc nonrec div_count([*]word divs) void:    word max, i, j;    max := dim(divs,1)-1;    divs[0] := 0;    for i from 1 upto max do divs[i] := 1 od;    for i from 2 upto max do        for j from i by i upto max do            divs[j] := divs[j] + 1        od    odcorp /* Find Tau numbers */proc nonrec main() void:    [1100]word divs;    word n, seen;     div_count(divs);    seen := 0;    n := 0;     while n := n + 1; seen < 100 do        if n % divs[n] = 0 then            seen := seen + 1;            write(n:5);            if seen % 10 = 0 then writeln() fi        fi    odcorp`
Output:
```    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 1016 1040 1044 1048 1056 1068 1089 1096```

## F#

` // Tau number. Nigel Galloway: March 9th., 2021Seq.initInfinite((+)1)|>Seq.filter(fun n->n%(tau n)=0)|>Seq.take 100|>Seq.iter(printf "%d "); printfn "" `
Output:
```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 1016 1040 1044 1048 1056 1068 1089 1096
```

## Factor

Works with: Factor version 0.99 2020-08-14
`USING: assocs grouping io kernel lists lists.lazy mathmath.functions math.primes.factors prettyprint sequencessequences.extras ; : tau ( n -- count ) group-factors values [ 1 + ] map-product ; : tau? ( n -- ? ) dup tau divisor? ; : taus ( -- list ) 1 lfrom [ tau? ] lfilter ; ! Task"The first 100 tau numbers are:" print100 taus ltake list>array 10 group simple-table. `
Output:
```The first 100 tau numbers are:
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 1016 1040 1044 1048 1056 1068 1089 1096
```

## Fermat

`Func Istau(t) =    if t<3 then Return(1) else        numdiv:=2;        for q = 2 to t\2 do            if Divides(q, t) then numdiv:=numdiv+1 fi;        od;        if Divides(numdiv, t)=1 then Return(1) else Return(0) fi;    fi;    .; numtau:=0;i:=0; while numtau<100 do    i:=i+1;    if Istau(i) = 1 then        numtau:=numtau+1;        !(i,'   ');        if Divides(10, numtau) then !! fi;    fi;od;`

## 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_tau_numbers ( n -- )  ." The first " dup . ." tau numbers are:" cr  0 >r  1  begin    over [email protected] >  while    dup dup divisor_count mod 0= if      dup 6 .r      r> 1+      dup 10 mod 0= if cr else space then      >r    then    1+  repeat  2drop rdrop ; 100 print_tau_numbersbye`
Output:
```The first 100 tau numbers are:
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   1016   1040   1044   1048   1056   1068   1089   1096
```

## FreeBASIC

`function numdiv( n as uinteger ) as uinteger    dim as uinteger c = 2    for i as uinteger = 2 to (n+1)\2        if n mod i = 0 then c += 1    next i    return cend function function istau( n as uinteger ) as boolean    if n = 1 then return true    if n mod numdiv(n) = 0 then return true else return falseend function dim as uinteger c = 0, i=1while c < 100    if istau(i) then        print i,        c += 1        if c mod 10 = 0 then print    end if    i += 1wend`
Output:
```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           1016          1040          1044          1048          1056          1068          1089          1096```

## Frink

`tau = {|x| x mod length[allFactors[x]] == 0}println[formatTable[columnize[first[select[count[1], tau], 100], 10], "right"]]`
Output:
```  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 1016 1040 1044 1048 1056 1068 1089 1096
```

## 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 first 100 tau numbers are:")    count := 0    i := 1    for count < 100 {        tf := countDivisors(i)        if i%tf == 0 {            fmt.Printf("%4d  ", i)            count++            if count%10 == 0 {                fmt.Println()            }        }        i++    }}`
Output:
```The first 100 tau numbers are:
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  1016  1040  1044  1048  1056  1068  1089  1096
```

`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) isTau :: Integral a => a -> BoolisTau n = 0 == mod n (tau n) main = print . take 100 . filter isTau \$ [1..]`
Output:
`[1,2,8,9,12,18,24,36,40,56,60,72,80,84,88,96,104,108,128,132,136,152,156,180,184,204,225,228,232,240,248,252,276,288,296,328,344,348,360,372,376,384,396,424,441,444,448,450,468,472,480,488,492,504,516,536,560,564,568,584,600,612,625,632,636,640,664,672,684,708,712,720,732,776,792,804,808,824,828,852,856,864,872,876,880,882,896,904,936,948,972,996,1016,1040,1044,1048,1056,1068,1089,1096]`

and we could also define Tau numbers in terms of a more general divisors function:

`import Data.List (group, scanl)import Data.List.Split (chunksOf)import Data.Numbers.Primes (primeFactors) ----------------------- TAU NUMBERS ---------------------- tauNumbers :: [Int]tauNumbers =  filter    ((0 ==) . (rem <*> (length . divisors)))    [1 ..] --------------------------- TEST -------------------------main :: IO ()main =  let xs = take 100 \$ fmap show tauNumbers      w = length \$ last xs   in (putStrLn . unlines) \$        unwords . fmap (justifyRight w ' ')          <\$> chunksOf 10 xs ------------------------- GENERIC ------------------------ divisors :: Int -> [Int]divisors =  foldr    (flip ((<*>) . fmap (*)) . scanl (*) 1)    [1]    . group    . primeFactors justifyRight :: Int -> Char -> String -> StringjustifyRight n c = (drop . length) <*> (replicate n c <>)`
Output:
```   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 1016 1040 1044 1048 1056 1068 1089 1096```

## J

Implementation:

`    tau_number=: 0 = (|~ [email protected]>)   tally_factors=: [: */ 1 + _&q: `

Explanation: _ q: produces a list of the exponents of the prime factors of a number. The product of 1 + this list is the number of positive factors of that number. We have a tau number if the remainder of the number divided by that factor count is zero.

In the task example, we generate a list of the first 2000 positive integers and then use an expression of the form (#~ test) which filters a list of numbers based on that test. We then extract the first 100 of these in a 4 row 25 column table.

`    (i.4 25){ (#~ tau_number) 1+i.2000  1   2   8   9  12  18  24  36  40  56  60  72  80  84  88  96 104  108  128  132  136  152  156  180  184204 225 228 232 240 248 252 276 288 296 328 344 348 360 372 376 384  396  424  441  444  448  450  468  472480 488 492 504 516 536 560 564 568 584 600 612 625 632 636 640 664  672  684  708  712  720  732  776  792804 808 824 828 852 856 864 872 876 880 882 896 904 936 948 972 996 1016 1040 1044 1048 1056 1068 1089 1096 `

## Java

Translation of: D
`public class Tau {    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 long limit = 100;        System.out.printf("The first %d tau numbers are:%n", limit);        long count = 0;        for (long n = 1; count < limit; ++n) {            if (n % divisorCount(n) == 0) {                System.out.printf("%6d", n);                ++count;                if (count % 10 == 0) {                    System.out.println();                }            }        }    }}`
Output:
```The first 100 tau numbers are:
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  1016  1040  1044  1048  1056  1068  1089  1096```

## 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] + .;`

`def taus: range(1;infinite) | select(. % count(divisors) == 0); # The first 100 Tau numbers:[limit(100; taus)]| nwise(10) | map(lpad(4)) | join(" ")`
Output:
```   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 1016 1040 1044 1048 1056 1068 1089 1096
```

## 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 function taunumbers(toget = 100)    n = 0    for i in 1:100000000        if i % numfactors(i) == 0            n += 1            print(rpad(i, 5), n % 20 == 0 ? " \n" : "")            n == toget && break        end    endend taunumbers() `
Output:
```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  1016 1040 1044 1048 1056 1068 1089 1096
```

## Lua

Translation of: C
`function divisor_count(n)    local total = 1     -- Deal with powers of 2 first    while (n & 1) == 0 do        total = total + 1        n = n >> 1    end    -- Odd prime factors up to the square root    local p = 3    while p * p <= n do        local count = 1        while n % p == 0 do            count = count + 1            n = math.floor(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 local limit = 100local count = 0print("The first " .. limit .. " tau numbers are:")local n = 1while count < limit do    if n % divisor_count(n) == 0 then        io.write(string.format("%6d", n))        count = count + 1        if count % 10 == 0 then            print()        end    end    n = n + 1end`
Output:
```The first 100 tau numbers are:
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  1016  1040  1044  1048  1056  1068  1089  1096```

`            NORMAL MODE IS INTEGER             INTERNAL FUNCTION(N)                ENTRY TO POSDIV.                COUNT = 1                THROUGH DIV, FOR I=2, 1, I.G.NDIV             WHENEVER N/I*I.E.N, COUNT = COUNT+1                FUNCTION RETURN COUNT            END OF FUNCTION             SEEN=0            THROUGH TAU, FOR X=1, 1, SEEN.GE.100            DIVS=POSDIV.(X)            WHENEVER X/DIVS*DIVS.E.X                PRINT FORMAT NUM,X                SEEN = SEEN+1TAU         END OF CONDITIONAL             VECTOR VALUES NUM = \$I4*\$            END OF PROGRAM `
Output:
```   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
1016
1040
1044
1048
1056
1068
1089
1096```

## Mathematica/Wolfram Language

`Take[Select[Range[10000], Divisible[#, Length[Divisors[#]]] &], 100]`
Output:
`{1,2,8,9,12,18,24,36,40,56,60,72,80,84,88,96,104,108,128,132,136,152,156,180,184,204,225,228,232,240,248,252,276,288,296,328,344,348,360,372,376,384,396,424,441,444,448,450,468,472,480,488,492,504,516,536,560,564,568,584,600,612,625,632,636,640,664,672,684,708,712,720,732,776,792,804,808,824,828,852,856,864,872,876,880,882,896,904,936,948,972,996,1016,1040,1044,1048,1056,1068,1089,1096}`

## Modula-2

`MODULE TauNumbers;FROM InOut IMPORT WriteCard, WriteLn; CONST     MaxNum = 1100; (* enough to generate 100 Tau numbers *)    NumTau = 100;  (* how many Tau numbers to generate *) VAR DivCount: ARRAY [1..MaxNum] OF CARDINAL;    seen, n: CARDINAL; (* Find the amount of divisors for each number beforehand *)PROCEDURE CountDivisors;VAR i, j: CARDINAL;BEGIN    FOR i := 1 TO MaxNum DO        DivCount[i] := 1; (* every number is divisible by 1 *)    END;     FOR i := 2 TO MaxNum DO        j := i;        WHILE j <= MaxNum DO (* J is divisible by I *)            DivCount[j] := DivCount[j] + 1;            j := j + i; (* next multiple of i *)        END;    END;END CountDivisors; BEGIN    CountDivisors();    n := 1;    seen := 0;    WHILE seen < NumTau DO        IF n MOD DivCount[n] = 0 THEN            WriteCard(n, 5);            INC(seen);            IF seen MOD 10 = 0 THEN                WriteLn();            END;        END;        INC(n);    END;END TauNumbers.`
Output:
```    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 1016 1040 1044 1048 1056 1068 1089 1096```

## 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 var count = 0var n = 1var tauNumbers: seq[Natural]while true:  if n mod divcount(n) == 0:    tauNumbers.add n    inc count    if count == 100: break  inc n echo "First 100 tau numbers:"for i, n in tauNumbers:  stdout.write (\$n).align(5)  if i mod 20 == 19: echo()`
Output:
```First 100 tau numbers:
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 1016 1040 1044 1048 1056 1068 1089 1096```

## Pascal

### Free Pascal

`program Tau_number;{\$IFDEF Windows}  {\$APPTYPE CONSOLE} {\$ENDIF}  function CountDivisors(n: NativeUint): integer;  //tau function  var    q, p, cnt, divcnt: NativeUint;  begin    divCnt := 1;    if n > 1 then    begin      cnt := 1;      while not (Odd(n)) do      begin        n := n shr 1;        divCnt+= cnt;      end;      p := 3;      while p * p <= n do      begin        cnt := divCnt;        q := n div p;        while q * p = n do        begin          n := q;          q := n div p;          divCnt+= cnt;        end;        Inc(p, 2);      end;      if n <> 1 then        divCnt += divCnt;    end;    CountDivisors := divCnt;  end; const  UPPERLIMIT = 100;var  cnt,n: NativeUint;begin  cnt := 0;  n := 1;  repeat    if n MOD CountDivisors(n) = 0 then    Begin      write(n:5);      inc(cnt);      if cnt Mod 10 = 0 then        writeln;    end;    inc(n);  until cnt >= UPPERLIMIT;  writeln;  {\$Ifdef Windows}readln;{\$ENDIF}end.`
TIO.RUN:
```    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 1016 1040 1044 1048 1056 1068 1089 1096
```

## Perl

Library: ntheory
`use strict;use warnings;use feature 'say';use ntheory 'divisors'; my(@x,\$n); do { push(@x,\$n) unless \$n % scalar(divisors(++\$n)) } until 100 == @x; say "Tau numbers - first 100:\n" .    ((sprintf "@{['%5d' x 100]}", @x[0..100-1]) =~ s/(.{80})/\$1\n/gr);`
Output:
```    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 1016 1040 1044 1048
1056 1068 1089 1096```

## Phix

### imperative

```integer n = 1, found = 0
while found<100 do
if remainder(n,length(factors(n,1)))=0 then
found += 1
printf(1,"%,6d",n)
if remainder(found,10)=0 then puts(1,"\n") end if
end if
n += 1
end while
```
Output:
```     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
```

### functional/memoised

same output

```sequence tau_cache = {1}
function tau(integer n)
while n>length(tau_cache) do
integer nt = tau_cache[\$]+1
while remainder(nt,length(factors(nt,1)))!=0 do
nt += 1
end while
tau_cache &= nt
end while
return tau_cache[n]
end function

puts(1,join_by(apply(true,sprintf,{{"%,6d"},apply(tagset(100),tau)}),1,10,""))
```

## PILOT

`T :1C :n=2C :seen=1C :max=100*numberC :c=1C :i=1*divisorC (n=i*(n/i)):c=c+1C :i=i+1J (i<=n/2):*divisorT (n=c*(n/c)):#nC (n=c*(n/c)):seen=seen+1C :n=n+1J (seen<max):*numberE :`
Output:
```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
1016
1040
1044
1048
1056
1068
1089
1096```

## PL/M

`100H:BDOS: PROCEDURE (FN, ARG); DECLARE FN BYTE, ARG ADDRESS; GO TO 5; END BDOS;EXIT: PROCEDURE; CALL BDOS(0,0); END EXIT;PRINT: PROCEDURE (S); DECLARE S ADDRESS; CALL BDOS(9,S); END PRINT; /* PRINT NUMBER RIGHT-ALIGNED IN 7 POSITIONS */PRINT\$NUMBER: PROCEDURE (N);    DECLARE S (7) BYTE INITIAL (' .....\$');    DECLARE N ADDRESS, I BYTE;    I = 6;DIGIT:    I = I - 1;    S(I) = N MOD 10 + '0';    N = N / 10;    IF N > 0 THEN GO TO DIGIT;    DO WHILE I <> 0;        I = I - 1;        S(I) = ' ';    END;    CALL PRINT(.S);END PRINT\$NUMBER; /* COUNT AND STORE AMOUNT OF DIVISORS FOR 1..N AT VEC */COUNT\$DIVS: PROCEDURE (VEC, N);    DECLARE (VEC, N, V BASED VEC) ADDRESS;    DECLARE (I, J) ADDRESS;     DO I=1 TO N;        V(I) = 1;    END;     DO I=2 TO N;        J = I;        DO WHILE J <= N;            V(J) = V(J) + 1;            J = J + I;        END;    END;END COUNT\$DIVS; /* GIVEN VECTOR OF COUNT OF DIVISORS, SEE IF N IS A TAU NUMBER */TAU: PROCEDURE (VEC, N) BYTE;    DECLARE (VEC, N, V BASED VEC) ADDRESS;    RETURN N MOD V(N) = 0;END TAU; DECLARE AMOUNT LITERALLY '100';DECLARE LIMIT LITERALLY '1100'; DECLARE SEEN BYTE INITIAL (0);DECLARE N ADDRESS INITIAL (1); CALL COUNT\$DIVS(.MEMORY, LIMIT);DO WHILE SEEN < AMOUNT;    IF TAU(.MEMORY, N) THEN DO;        CALL PRINT\$NUMBER(N);        SEEN = SEEN + 1;        IF SEEN MOD 10 = 0 THEN CALL PRINT(.(13,10,'\$'));    END;    N = N + 1;END; CALL EXIT;EOF`
Output:
```     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  1016  1040  1044  1048  1056  1068  1089  1096```

## PureBasic

Translation of: FreeBasic
`OpenConsole() Procedure.i numdiv(n)  c=2  For i=2 To (n+1)/2 : If n%i=0 : c+1 : EndIf : Next  ProcedureReturn cEndProcedure Procedure.b istau(n)  If n=1 : ProcedureReturn #True : EndIf  If n%numdiv(n)=0 : ProcedureReturn #True : Else : ProcedureReturn #False : EndIfEndProcedure c=0 : i=1While c<100  If istau(i) : Print(RSet(Str(i),4)+#TAB\$) : c+1 : If c%10=0 : PrintN("") : EndIf: EndIf  i+1Wend Input()`
Output:
```   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	1016	1040	1044	1048	1056	1068	1089	1096	```

## Python

### Python: Procedural

`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 def is_tau_number(n):    assert(isinstance(n, int))    if n <= 0:        return False    return 0 == n%tau(n) if __name__ == "__main__":    n = 1    ans = []    while len(ans) < 100:        if is_tau_number(n):            ans.append(n)        n += 1    print(ans)`
Output:
`[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, 1016, 1040, 1044, 1048, 1056, 1068, 1089, 1096]`

### Python: Functional

Composing pure functions, and defining a non-finite stream of Tau numbers in terms of a generic `divisors` function:

`'''Tau numbers''' from operator import mulfrom math import floor, sqrtfrom functools import reducefrom itertools import (    accumulate, chain, count,    groupby, islice, product)  # tauNumbers :: Generator [Int]def tauNumbers():    '''Positive integers divisible by the       count of their positive divisors.    '''    return (        n for n in count(1)        if 0 == n % len(divisors(n))    )  # ------------------------- TEST -------------------------# main :: IO ()def main():    '''The first hundred Tau numbers.    '''    xs = take(100)(        tauNumbers()    )    w = len(str(xs[-1]))    print('\n'.join([        ' '.join([            str(cell).rjust(w, ' ') for cell in row        ])        for row in chunksOf(10)(xs)    ]))  # ----------------------- 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):    '''The ordered divisors of n.    '''    def go(a, x):        return [a * b for a, b in product(            a,            accumulate(chain([1], x), mul)        )]    return sorted(        reduce(go, [            list(g) for _, g            in groupby(primeFactors(n))        ], [1])    ) if 1 < n else [1]  # primeFactors :: Int -> [Int]def primeFactors(n):    '''A list of the prime factors of n.    '''    def f(qr):        r = qr[1]        return step(r), 1 + r     def step(x):        return 1 + (x << 2) - ((x >> 1) << 1)     def go(x):        root = floor(sqrt(x))         def p(qr):            q = qr[0]            return root < q or 0 == (x % q)         q = until(p)(f)(            (2 if 0 == x % 2 else 3, 1)        )[0]        return [x] if q > root else [q] + go(x // q)     return go(n)  # 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  # until :: (a -> Bool) -> (a -> a) -> a -> adef until(p):    '''The result of repeatedly applying f until p holds.       The initial seed value is x.    '''    def go(f):        def g(x):            v = x            while not p(v):                v = f(v)            return v        return g    return go  # MAIN ---if __name__ == '__main__':    main() `
Output:
```   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 1016 1040 1044 1048 1056 1068 1089 1096```

## Quackery

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

`  [ dup factors size mod 0 = ] is taunumber ( n --> b )   [] 0  [ 1+ dup taunumber if      [ tuck join swap ]    over size 100 = until ]  drop  [] swap   witheach [ number\$ nested join ]  80 wrap\$`
Output:
```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 1016 1040 1044 1048 1056 1068 1089 1096```

## R

`tau <- function(t){  results <- integer(0)  resultsCount <- 0  n <- 1  while(resultsCount != t)  {    condition <- function(n) (n %% length(c(Filter(function(x) n %% x == 0, seq_len(n %/% 2)), n))) == 0    if(condition(n))    {      resultsCount <- resultsCount + 1      results[resultsCount] <- n    }    n <- n + 1  }  results}tau(100)`

## 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 pgm displays   N   tau numbers,  an integer divisible by the # of its divisors). */parse arg n cols .                               /*obtain optional argument from the CL.*/if    n=='' |    n==","  then    n= 100          /*Not specified?  Then use the default.*/if cols=='' | cols==","  then cols=  10          /*Not specified?  Then use the default.*/w= max(8, length(n) )                            /*W:  used to align 1st output column. */@tau= ' the first '  commas(n)   " tau numbers " /*the title of the tau numbers table.  */say ' index │'center(@tau,  1 + cols*(w+1)     ) /*display the title of the output table*/say '───────┼'center(""  ,  1 + cols*(w+1), '─') /*   "     " header  "  "     "     "  */idx= 1;                  #= 0;           \$=      /*idx: line;   #:  tau numbers;  \$: #s */           do j=1  until #==n                    /*search for   N   tau numbers         */           if j//tau(j) \==0  then iterate       /*Is this a tau number?  No, then skip.*/           #= # + 1                              /*bump the count of tau numbers found. */           \$= \$  right( commas(j), w)            /*add a tau number to the output list. */           if #//cols\==0     then iterate       /*Not a multiple of cols?  Don't show. */           say center(idx, 7)'│'   substr(\$, 2)  /*display partial list to the terminal.*/           idx= idx + cols;               \$=     /*bump idx by number of cols; nullify \$*/           end   /*j*/ if \$\==''  then say center(idx, 7)"│"  substr(\$, 2)  /*possible display residual output.*/say '───────┴'center(""  ,  1 + cols*(w+1), '─')exit 0                                           /*stick a fork in it,  we're all done. *//*──────────────────────────────────────────────────────────────────────────────────────*/commas: parse arg ?;  do jc=length(?)-3  to 1  by -3; ?=insert(',', ?, jc); end;  return ?/*──────────────────────────────────────────────────────────────────────────────────────*/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 do;   #= 2;               end  /*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*/                              /* [↑]  this form of DO loop is faster.*/     return #`
output   when using the default input:
``` index │                                the first  100  tau numbers
───────┼───────────────────────────────────────────────────────────────────────────────────────────
1   │        1        2        8        9       12       18       24       36       40       56
11   │       60       72       80       84       88       96      104      108      128      132
21   │      136      152      156      180      184      204      225      228      232      240
31   │      248      252      276      288      296      328      344      348      360      372
41   │      376      384      396      424      441      444      448      450      468      472
51   │      480      488      492      504      516      536      560      564      568      584
61   │      600      612      625      632      636      640      664      672      684      708
71   │      712      720      732      776      792      804      808      824      828      852
81   │      856      864      872      876      880      882      896      904      936      948
91   │      972      996    1,016    1,040    1,044    1,048    1,056    1,068    1,089    1,096
───────┴───────────────────────────────────────────────────────────────────────────────────────────
```

## Ring

` see "The first 100 tau numbers are:" + nl + nl n = 1num = 0limit = 100while num < limit      n = n + 1      tau = 0      for m = 1 to n          if n%m = 0             tau = tau + 1          ok      next      if n%tau = 0         num = num + 1         if num%10 = 1            see nl         ok         see "" + n + " "      okend `

Output:

```The first 100 tau numbers are:

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 1016 1040 1044 1048 1056 1068 1089 1096
```

## Ruby

`require 'prime' taus = Enumerator.new do |y|  (1..).each do |n|    num_divisors = n.prime_division.inject(1){|prod, n| prod *= n[1] + 1 }    y << n if n % num_divisors == 0  endend p taus.take(100) `
Output:
```[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, 1016, 1040, 1044, 1048, 1056, 1068, 1089, 1096]
```

## Rust

` /// Gets all divisors of a number, including itselffn get_divisors(n: u32) -> Vec<u32> {    let mut results = Vec::new();     for i in 1..(n / 2 + 1) {        if n % i == 0 {            results.push(i);        }    }    results.push(n);    results} fn is_tau_number(i: u32) -> bool {    0 == i % get_divisors(i).len() as u32} fn main() {    println!("\nFirst 100 Tau numbers:");    let mut counter: u32 = 0;    let mut i: u32 = 1;    while counter < 100 {        if is_tau_number(i) {            print!("{:>4}", i);            counter += 1;            print!("{}", if counter % 20 == 0 { "\n" } else { "," });        }        i += 1;    }}  `
Output:
```First 100 Tau numbers:
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, 996,1016,1040,1044,1048,1056,1068,1089,1096
```

## Sidef

`func is_tau_number(n) {    n % n.sigma0 == 0} say is_tau_number.first(100).join(' ')`
Output:
```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 1016 1040 1044 1048 1056 1068 1089 1096
```

## 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("The first \(limit) tau numbers are:")var count = 0var n = 1while count < limit {    if n % divisorCount(number: n) == 0 {        print(String(format: "%5d", n), terminator: "")        count += 1        if count % 10 == 0 {            print()        }    }    n += 1}`
Output:
```The first 100 tau numbers are:
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 1016 1040 1044 1048 1056 1068 1089 1096
```

## Verilog

`module main;  integer n, m, num, limit, tau;   initial begin    \$display("The first 100 tau numbers are:\n");    n = 0;    num = 0;    limit = 100;     while (num < limit) begin      n = n + 1;      tau = 0;      for (m = 1; m <= n; m=m+1) if (n % m == 0) tau = tau + 1;       if (n % tau == 0) begin        num = num + 1;        if (num % 5 == 1) \$display("");        \$write(n);      end    end    \$finish ;  endendmodule`

## VTL-2

`10 N=110020 I=130 :I)=140 I=I+150 #=N>I*3060 I=270 J=I80 :J)=:J)+190 J=J+I100 #=N>J*80110 I=I+1120 #=N>I*70130 C=0140 I=1150 #=I/:I)*0+0<%*210160 ?=I170 \$=9180 C=C+1190 #=C/10*0+0<%*210200 ?=""210 I=I+1220 #=C<100*150`
Output:
```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     1016    1040    1044    1048    1056    1068    1089    1096```

## Wren

Library: Wren-math
Library: Wren-fmt
`import "/math" for Intimport "/fmt" for Fmt System.print("The first 100 tau numbers are:")var count = 0var i = 1while (count < 100) {    var tf = Int.divisors(i).count    if (i % tf == 0) {        Fmt.write("\$,5d  ", i)        count = count + 1        if (count % 10 == 0) System.print()    }    i = i + 1}`
Output:
```The first 100 tau numbers are:
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
```

## XPL0

`func Divs(N);   \Return number of divisors of Nint  N, D, C;[C:= 0;for D:= 1 to N do    if rem(N/D) = 0 then C:= C+1;return C;]; int C, N;[Format(5, 0);C:= 0;  N:= 1;loop    [if rem(N/Divs(N)) = 0 then            [RlOut(0, float(N));            C:= C+1;            if rem(C/10) = 0 then CrLf(0);            if C >= 100 then quit;            ];        N:= N+1;        ];]`
Output:
```    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 1016 1040 1044 1048 1056 1068 1089 1096
```