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


(de mdr-mp (N)
   "Returns the solutions in a list, i.e., '(MDR MP)"
   (let MP 0
      (while (< 1 (length N))
         (setq N (apply * (mapcar format (chop N))))
         (inc 'MP) )
      (list N MP) ) )
 
 
 
# Get the MDR/MP of these nums.
(setq Test-nums '(123321 7739 893 899998))
 
(let Fmt (6 5 5)
   (tab Fmt "Values" "MDR" "MP")
   (tab Fmt "======" "===" "==")
   (for I Test-nums
      (let MDR-MP (mdr-mp I)
         (tab Fmt I (car MDR-MP) (cadr MDR-MP)) ) ) )
 
(prinl)
 
# Get the nums of these MDRs.
(setq *Want 5)
 
(setq *Solutions (make (for MDR (range 0 9)
   (link (make (let N 0 (until (= *Want (length (made)))
      (when (= MDR (car (mdr-mp N)))
         (link N) )
      (inc 'N) )))) )))
 
(let Fmt (3 1 -1)
   (tab Fmt "MDR" ":  " "Values")
   (tab Fmt "===" "   " "======")
   (for (I . S) *Solutions
      (tab Fmt (dec I) ":  " (glue ", " S)) ) )