Tau number

From Rosetta Code
Task
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.


Task

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).


Related task



11l[edit]

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] ans
L 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![edit]

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
  FI
RETURN (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
  OD
RETURN
Output:

Screenshot from Atari 8-bit computer

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[edit]

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
    END
END
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[edit]

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 do
begin
    j := i;
    while j <= 1100 do
    begin
        dcount[j] := dcount[j] + 1;
        j := j + i;
    end;
end;

n := 0;
i := 1;
while n < 100 do
begin
    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[edit]

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[edit]

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 counter
end factorCount

-- Task code:
local output, n, counter, astid
set output to {"First 100 tau numbers:"}
set n to 0
set counter to 0
repeat 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 if
end repeat
set astid to AppleScript's text item delimiters
set AppleScript's text item delimiters to ""
set output to output as text
set AppleScript's text item delimiters to astid
return 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[edit]

tau: function [x] -> size factors x

found: 0
i:1
while [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

AutoHotkey[edit]

n := c:= 0
while (c<100)
    if isTau(++n)
        c++, result .= SubStr("   " n, -3) . (Mod(c, 10) ? " " : "`n")
MsgBox % result
return

isTau(num){
    return (num/(n := StrSplit(Factors(num), ",").Count()) = floor(num/n))
}

Factors(n) {
    Loop, % floor(sqrt(n))
        v := A_Index = 1 ? 1 "," n : mod(n,A_Index) ? v : v "," A_Index "," n//A_Index
    Sort, v, N U D,
    Return, v
}
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[edit]

# syntax: GAWK -f TAU_NUMBER.AWK
BEGIN {
    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[edit]

10 DEFINT A-Z
20 S=0: N=1
30 C=1
40 IF N<>1 THEN FOR I=1 TO N/2: C=C-(N MOD I=0): NEXT
50 IF N MOD C=0 THEN PRINT N,: S=S+1
60 N=N+1
70 IF S<100 THEN 30
80 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[edit]

print "The first 100 tau numbers are:"

n = 0
num = 0
limit = 100
while 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 if
end while
end

QBasic[edit]

Works with: QBasic version 1.1
PRINT "The first 100 tau numbers are:"

n = 0
num = 0
limit = 100
DO
      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 IF
LOOP WHILE num < limit
END

True BASIC[edit]

LET n = 0
LET num = 0
LET limit = 100
DO
   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 IF
LOOP WHILE num < limit
END

Yabasic[edit]

print "The first 100 tau numbers are:"

n = 0
num = 0
limit = 100
while 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 if
wend
print
end


BCPL[edit]

get "libhdr"

// Count the divisors of 1..N
let 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[edit]

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++[edit]

#include <iomanip>
#include <iostream>

// See https://en.wikipedia.org/wiki/Divisor_function
unsigned 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[edit]

% 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 limit
tau_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
    end
end tau_numbers

% Show the first 100 Tau numbers
start_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
    end
end 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[edit]

include "cowgol.coh";

# Get count of positive divisors of number
sub 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 numbers
var 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[edit]

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[edit]

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[edit]

/* 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
    od
corp

/* 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
    od
corp
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#[edit]

This task uses [Tau_function#F.23]

// Tau number. Nigel Galloway: March 9th., 2021
Seq.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[edit]

Works with: Factor version 0.99 2020-08-14
USING: assocs grouping io kernel lists lists.lazy math
math.functions math.primes.factors prettyprint sequences
sequences.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:" print
100 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[edit]

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[edit]

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 r@ >
  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_numbers
bye
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[edit]

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 c
end 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 false
end function

dim as uinteger c = 0, i=1
while c < 100
    if istau(i) then
        print i,
        c += 1
        if c mod 10 = 0 then print
    end if
    i += 1
wend
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[edit]

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[edit]

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  

Haskell[edit]

tau :: Integral a => a -> a
tau 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 -> Bool
isTau 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 -> String
justifyRight 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[edit]

Implementation:

   tau_number=: 0 = (|~ tally_factors@>)
   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.

Task example:

   (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  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

Java[edit]

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[edit]

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-printing
def nwise($n):
  def n: if length <= $n then . else .[0:$n] , (.[$n:] | n) end;
  n;

def lpad($len): tostring | ($len - length) as $l | (" " * $l)[:$l] + .;

The task

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[edit]

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
    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  

Lua[edit]

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 total
end

local limit = 100
local count = 0
print("The first " .. limit .. " tau numbers are:")
local n = 1
while 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 + 1
end
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

MAD[edit]

            NORMAL MODE IS INTEGER
            
            INTERNAL FUNCTION(N)
                ENTRY TO POSDIV.
                COUNT = 1
                THROUGH DIV, FOR I=2, 1, I.G.N
DIV             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+1
TAU         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[edit]

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[edit]

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[edit]

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 = 0
var n = 1
var 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[edit]

Free Pascal[edit]

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[edit]

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[edit]

imperative[edit]

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[edit]

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[edit]

T :1
C :n=2
C :seen=1
C :max=100
*number
C :c=1
C :i=1
*divisor
C (n=i*(n/i)):c=c+1
C :i=i+1
J (i<=n/2):*divisor
T (n=c*(n/c)):#n
C (n=c*(n/c)):seen=seen+1
C :n=n+1
J (seen<max):*number
E :
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[edit]

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[edit]

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 c
EndProcedure

Procedure.b istau(n)
  If n=1 : ProcedureReturn #True : EndIf
  If n%numdiv(n)=0 : ProcedureReturn #True : Else : ProcedureReturn #False : EndIf
EndProcedure

c=0 : i=1
While c<100
  If istau(i) : Print(RSet(Str(i),4)+#TAB$) : c+1 : If c%10=0 : PrintN("") : EndIf: EndIf
  i+1
Wend

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[edit]

Python: Procedural[edit]

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[edit]

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

'''Tau numbers'''

from operator import mul
from math import floor, sqrt
from functools import reduce
from 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 -> String
def 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 -> a
def 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[edit]

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[edit]

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[edit]

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[edit]

/*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[edit]

see "The first 100 tau numbers are:" + nl + nl

n = 1
num = 0
limit = 100
while 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 + " "
      ok
end

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[edit]

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
  end
end

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[edit]

/// Gets all divisors of a number, including itself
fn 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[edit]

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[edit]

import Foundation

// See https://en.wikipedia.org/wiki/Divisor_function
func 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 = 100
print("The first \(limit) tau numbers are:")
var count = 0
var n = 1
while 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[edit]

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 ;
  end
endmodule


VTL-2[edit]

10 N=1100
20 I=1
30 :I)=1
40 I=I+1
50 #=N>I*30
60 I=2
70 J=I
80 :J)=:J)+1
90 J=J+I
100 #=N>J*80
110 I=I+1
120 #=N>I*70
130 C=0
140 I=1
150 #=I/:I)*0+0<%*210
160 ?=I
170 $=9
180 C=C+1
190 #=C/10*0+0<%*210
200 ?=""
210 I=I+1
220 #=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[edit]

Library: Wren-math
Library: Wren-fmt
import "/math" for Int
import "/fmt" for Fmt

System.print("The first 100 tau numbers are:")
var count = 0
var i = 1
while (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[edit]

func Divs(N);   \Return number of divisors of N
int  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