Primorial numbers

Primorial numbers is the number series formed by multiplying successive prime numbers.

The series is:

primorial(0) = 1 (by definition)
primorial(1) = 2 (2)
primorial(2) = 6 (2*3)
primorial(3) = 30 (2*3*5)
primorial(4) = 210 (2*3*5*7)
primorial(5) = 2310 (2*3*5*7*11)
primorial(6) = 30030 (2*3*5*7*11*13)
∙ ∙ ∙

To express this mathematically, primorial_n is the product of the first n (successive) primes:

primorial_{n}=\prod _{k=1}^{n}prime_{k}

where prime_k is the k^th prime number.

In some sense, generating primorial numbers is similar to factorials. As with factorials, primorial numbers get large quickly.

task requirements:

Show the first ten primorial numbers (0 ──► 9, inclusive).
Show the length of primorial numbers whose index is: 10 100 1,000 10,000 and 100,000.
Show the length of the one millionth primorial number (optionally).

links:

See the MathWorld webpage: primorial

See the Wikipedia webpage: primorial.

See the OEIS webpage: A2110.

REXX

<lang rexx>/*REXX program computes some primorial numbers for low #s, and for 10^n. */ parse arg N H . /*get optional arguments: N, L, H */ if N== | N==',' then N=10 /*was N given? Then use the default.*/ if H== | H==',' then H=100000 /*was H given? Then use the default.*/ numeric digits 600000; w=length(digits()) /* be able to handle large numbers.*/ @.=.; @.0=1; @.1=2; @.2=3; @.3=5; @.4=7; @.5=11; @.6=13 /*some low primes.*/

            s.1=4; s.2=9; s.3=25; s.4=49; s.5=121; s.6=169 /*squared primes. */

=6 /*number of primes*/

    do j=0  for N                     /*calculate the first  N  primorial #s.*/
    say right(j,length(N))    ' primorial is: '    primorial(j)
    end   /*j*/

p=1; say

    do k=1  for H                     /*process a large range of numbers.    */
    p=p*prime(k)                      /*calculate the next primorial number. */
    parse var k L 2  -1 R           /*get the left and rightmost dec digits*/
    if R\==0           then iterate   /*if right─most decimal digit\==0, skip*/
    if L\==1           then iterate   /* "  left─most    "      "  \==1,   " */
    if strip(k,,0)\==1 then iterate   /*Not a power of 10?  Then skip this K.*/
    say right(k,w) ' primorial length in decimal digits is:' right(length(p),w)
    end   /*k*/

exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────PRIMORIAL subroutine──────────────────────*/ primorial: procedure expose @. s. #; parse arg y; !=1 /*obtain the arg Y.*/

              do p=0  to y;  !=!*prime(p);  end   /*p*/  /*calculate product.*/

return ! /*return with the #.*/ /*──────────────────────────────────PRIME subroutine──────────────────────────*/ prime: procedure expose @. s. #; parse arg n; if @.n\==. then return @.n numeric digits 9 /*limit digs to min.*/

 do j=@.#+2  by 2                                        /*start looking at #*/
 if j//2==0  then iterate; if j//3==0    then iterate    /*divisible by 2│3 ?*/
 parse var j  -1 _;      if _==5       then iterate    /*right-most dig≡5? */
 if j//7==0  then iterate; if j//11==0   then iterate    /*divisible by 7│11?*/
   do k=6  while s.k<=j;   if j//@.k==0  then iterate j  /*divide by primes. */
   end   /*k*/
 #=#+1;  @.#=j;  s.#=j*j;  return j                      /*next prime; return*/
 end     /*j*/</lang>

output when using the default inputs:

 0  primorial is:  1
 1  primorial is:  2
 2  primorial is:  6
 3  primorial is:  30
 4  primorial is:  210
 5  primorial is:  2310
 6  primorial is:  30030
 7  primorial is:  510510
 8  primorial is:  9699690
 9  primorial is:  223092870

    10  primorial length in decimal digits is:     10
   100  primorial length in decimal digits is:    220
  1000  primorial length in decimal digits is:   3393
 10000  primorial length in decimal digits is:  45337
100000  primorial length in decimal digits is: 563921