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


 
(defun mdr/p (n)
  "Return a list with MDR and MP of n"
  (if (< n 10) 
    (list n 0)
    (mdr/p-aux n 1 1)))
 
(defun mdr/p-aux (n a c)
  (cond ((and (zerop n) (< a 10)) (list a c))
	((zerop n) (mdr/p-aux a 1 (+ c 1)))
	(t (mdr/p-aux (floor n 10) (* (rem n 10) a) c))))
 
(defun first-n-number-for-each-root (n &optional (r 0) (lst nil) (c 0))
  "Return the first m number with MDR = 0 to 9"
  (cond ((and (= (length lst) n) (= r 9)) (format t "~3@a: ~a~%" r (reverse lst)))
	((= (length lst) n) (format t "~3@a: ~a~%" r (reverse lst))
	                    (first-n-number-for-each-root n (+ r 1) nil 0))
	((= (first (mdr/p c)) r) (first-n-number-for-each-root n r (cons c lst) (+ c 1)))
	(t (first-n-number-for-each-root n r lst (+ c 1)))))
 
(defun start ()
  (format t "Number: MDR  MD~%")
  (loop for el in '(123321 7739 893 899998)
        do (format t "~6@a: ~{~3@a ~}~%" el (mdr/p el)))
  (format t "~%MDR: [n0..n4]~%")
  (first-n-number-for-each-root 5))