Tau number

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

Task

Show the first 100 Tau numbers.

Related task

Tau function

REXX

<lang rexx>/*REXX program counts the number of divisors (tau, or sigma_0) up to and including N.*/ parse arg n . /*obtain optional argument from the CL.*/ if n== | n=="," then n= 100 /*Not specified? Then use the default.*/ say 'the number of divisors (tau) for integers up to ' n " (inclusive):"; say say '─index─' center(" tau (number of divisors) ", 80, '─') w= max(7, length(n) ) /*W: used to align 1st output column. */ $= /*$: the output list, shown 20/line. */

               do j=1  for n                    /*list # proper divisors (tau) 1 ──► N */
               $= $  ||  right( tau(j), 4)      /*add a tau number to the output list. */
               if j//20\==0  then iterate       /*Not a multiple of 20?  Don't display.*/
               say center(j-19, 7)  $;  $=      /*display partial list to the terminal.*/
               end   /*j*/

if $\== then say center(j-1, 7) $ /*any residuals left to display ? */ exit 0 /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ tau: procedure; parse arg x 1 y /*X and $ are both set from the arg.*/

    if x<6  then return 2 + (x==4) - (x==1)     /*some low #s should be handled special*/
    odd= x // 2                                 /*check if  X  is odd (remainder of 1).*/
    if odd  then 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.*/</lang>

output when using the default input:

the number of divisors  (tau)  for integers up to  100  (inclusive):

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