The multiplicative digital root (MDR) and multiplicative persistence (MP) of a number, 



n


{\displaystyle n}

, is calculated rather like the Digital root except digits are multiplied instead of being added:

 Set 



m


{\displaystyle m}

 to 



n


{\displaystyle n}

 and 



i


{\displaystyle i}

 to 



0


{\displaystyle 0}

.
 While 



m


{\displaystyle m}

 has more than one digit:
 Find a replacement 



m


{\displaystyle m}

 as the multiplication of the digits of the current value of 



m


{\displaystyle m}

.
 Increment 



i


{\displaystyle i}

.
 Return 



i


{\displaystyle i}

 (= MP) and 



m


{\displaystyle m}

 (= MDR)


Task

 Tabulate the MP and MDR of the numbers 123321, 7739, 893, 899998
 Tabulate MDR versus the first five numbers having that MDR, something like:
MDR: [n0..n4]
===  ========
  0: [0, 10, 20, 25, 30]
  1: [1, 11, 111, 1111, 11111]
  2: [2, 12, 21, 26, 34]
  3: [3, 13, 31, 113, 131]
  4: [4, 14, 22, 27, 39]
  5: [5, 15, 35, 51, 53]
  6: [6, 16, 23, 28, 32]
  7: [7, 17, 71, 117, 171]
  8: [8, 18, 24, 29, 36]
  9: [9, 19, 33, 91, 119]

Show all output on this page.

Similar

The Product of decimal digits of n page was redirected here, and had the following description


Find the product of the decimal digits of a positive integer   n,   where n <= 100

The three existing entries for Phix, REXX, and Ring have been moved here, under ===Similar=== headings, feel free to match or ignore them.



References

 Multiplicative Digital Root on Wolfram Mathworld.
 Multiplicative digital root on The On-Line Encyclopedia of Integer Sequences.
 What's special about 277777788888899? - Numberphile video


(
& ( MP/MDR
  =   prod L n
    .   ( prod
        =   d
          .   @(!arg:%@?d ?arg)&!d*prod$!arg
            | 1
        )
      & !arg:?L
      &   whl
        ' ( @(!arg:? [>1)
          & (prod$!arg:?arg) !L:?L
          )
      & !L:? [?n
      & (!n+-1.!arg)
  )
& ( test
  =   n
    .     !arg:%?n ?arg
        & out$(!n "\t:" MP/MDR$!n)
        & test$!arg
      | 
  )
& test$(123321 7739 893 899998)
& 0:?i
& 1:?collecting:?done
&   whl
  ' ( !i+1:?i
    & MP/MDR$!i:(?MP.?MDR)
    & ( !done:?*(!MDR.)^((?.)+?)*?
      |   (!MDR.)^(!i.)*!collecting:?collecting
        & (   !collecting:?A*(!MDR.)^(?is+[5)*?Z
            & !A*!Z:?collecting
            & (!MDR.)^!is*!done:?done
          | 
          )
      )
    & !collecting:~1
    )
&   whl
  ' ( !done:(?MDR.)^?is*?done
    & put$(!MDR ":")
    & whl'(!is:(?i.)+?is&put$(!i " "))
    & put$\n
    )
);