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


Red ["Multiplicative digital root"]
 
mdr: function [
    "Returns a block containing the mdr and persistence of an integer"
    n [integer!]
][
    persistence: 0
    while [n > 10][
        product: 1
        m: n
        while [m > 0][
            product: m % 10 * product
            m: to-integer m / 10
        ]
        persistence: persistence + 1
        n: product
    ]
    reduce [n persistence]
]
 
foreach n [123321 7739 893 899998][
    result: mdr n
    print [pad n 6 "has multiplicative persistence" result/2 "and MDR" result/1]
]
 
print [newline "First five numbers with MDR of"]
 
repeat i 10 [
    prin rejoin [i - 1 ": "]
    hits: n: 0
    while [hits < 5][
        if i - 1 = first mdr n [
            prin pad n 5
            hits: hits + 1
        ]
        n: n + 1
    ]
    prin newline
]