Numbers whose count of divisors is prime

Numbers whose count of divisors is prime is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

Task

Find numbers which count of divisors is prime but not equal to 2, where n < 1,000

REXX

<lang rexx>/*REXX program finds positive integers N whose count of divisiors is prime, where N<200.*/ parse arg hi cols . /*obtain optional arguments from the CL*/ if hi== | hi=="," then hi= 1000 /*Not specified? Then use the defaults*/ if cols== | cols=="," then cols= 10 /* " " " " " " */ call genP /*build array of semaphores for primes.*/ w= 10 /*W: the maximum width of any column. */ title= ' positive integers N whose number of divisors is prime, where N < ' hi say ' index │'center(title, 1 + cols*(w+1) ) say '───────┼'center("" , 1 + cols*(w+1), '─') finds= 0; idx= 1; $=

        do j=1  for hi-1                        /*process positive integers in range.  */
        if !.j  then iterate                    /*Is  J  prime?  Then skip this number.*/
                else do;  n= nDivs(j)           /*get number of divisors of composite J*/
                          if n==2  then iterate
                          if \!.n  then iterate /*Number divisors prime?  No, then skip*/
                     end
        finds= finds + 1                        /*bump the number of found numbers.    */
        $= $  right(j, w)                       /*add a positive integer  ──►  $ list. */
        if finds//cols\==0 then iterate         /*have we populated a line of output?  */
        say center(idx, 7)'│'  substr($, 2); $= /*display what we have so far  (cols). */
        idx= idx + cols                         /*bump the  index  count for the output*/
        end   /*j*/                             /* [↑]   process a range of integers.  */

if $\== then say center(idx, 7)"│" substr($, 2) /*possible display residual output.*/ say '───────┴'center("" , 1 + cols*(w+1), '─') say say 'Found ' finds title exit 0 /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ nDivs: procedure; parse arg x; d= 2; if x==1 then return 1 /*handle special case of 1*/

      odd= x // 2                               /* [↓]  process EVEN or ODD ints.   ___*/
             do j=2+odd  by 1+odd  while j*j<x  /*divide by all the integers up to √ x */
             if x//j==0  then d= d + 2          /*÷?  Add two divisors to the total.   */
             end   /*j*/                        /* [↑]  %  ≡  integer division.        */
      if j*j==x  then  return  d + 1            /*Was X a square?  Then add 1 to total.*/
                       return  d                /*return the total.                    */

/*──────────────────────────────────────────────────────────────────────────────────────*/ genP: @.1=2; @.2=3; @.3=5; @.4=7; @.5=11 /*define some low primes. */

     !.=0;  !.2=1; !.3=1; !.5=1; !.7=1;  !.11=1 /*   "     "   "    "     semaphores.  */
                          #=5;   s.#= @.# **2   /*number of primes so far;     prime². */
       do j=@.#+2  by 2  to hi-1                /*find odd primes from here on.        */
       parse var j  -1 _; if     _==5  then iterate  /*J divisible by 5?  (right dig)*/
                            if j// 3==0  then iterate  /*"     "      " 3?             */
                            if j// 7==0  then iterate  /*"     "      " 7?             */
              do k=5  while s.k<=j              /* [↓]  divide by the known odd primes.*/
              if j // @.k == 0  then iterate j  /*Is  J ÷ X?  Then not prime.     ___  */
              end   /*k*/                       /* [↑]  only process numbers  ≤  √ J   */
       #= #+1;    @.#= j;    s.#= j*j;   !.j= 1 /*bump # of Ps; assign next P;  P²; P# */
       end          /*j*/;   return</lang>

output   when using the default inputs:

 index │                   positive integers  N  whose number of divisors is prime,  where  N <  1000
───────┼───────────────────────────────────────────────────────────────────────────────────────────────────────────────
   1   │          4          9         16         25         49         64         81        121        169        289
  11   │        361        529        625        729        841        961
───────┴───────────────────────────────────────────────────────────────────────────────────────────────────────────────

Found  16  positive integers  N  whose number of divisors is prime,  where  N <  1000

Ring

<lang ring> load "stdlib.ring" row = 0

see "working..." + nl see "Numbers which count of divisors is prime are:" + nl

for n = 1 to 1000

   num = 0
   for m = 1 to n
       if n%m = 0
          num++
       ok
   next
   if isprime(num) and num != 2
      see "" + n + " "
      row++
      if row%5 = 0
         see nl
      ok
   ok  

next

see nl + "Found " + row + " numbers" + nl see "done..." + nl </lang>

Output:

working...
Numbers which count of divisors is prime are:
4 9 16 25 49 
64 81 121 169 289 
361 529 625 729 841 
961 
Found 16 numbers
done...