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


' FB 1.05.0 Win64
 
Function multDigitalRoot(n As UInteger, ByRef mp As Integer, base_ As Integer = 10) As Integer
  Dim mdr As Integer
  mp = 0
  Do 
    mdr = IIf(n > 0, 1, 0) 
    While n > 0
      mdr *= n Mod base_
      n = n \ base_
    Wend
    mp += 1
    n = mdr
  Loop until mdr < base_
  Return mdr
End Function
 
Dim As Integer mdr, mp
Dim a(3) As UInteger = {123321, 7739, 893, 899998}
For i As UInteger = 0 To 3
 mp = 0
 mdr = multDigitalRoot(a(i), mp)
 Print a(i); Tab(10); "MDR ="; mdr; Tab(20); "MP ="; mp
 Print
Next
Print
Print "MDR    1     2     3     4     5"
Print "===  ==========================="
Print
Dim num(0 To 9, 0 To 5) As UInteger '' all zero by default
Dim As UInteger n = 0, count = 0
Do
  mdr = multDigitalRoot(n, mp)
  If num(mdr, 0) < 5 Then
    num(mdr, 0) += 1
    num(mdr, num(mdr, 0)) = n
    count += 1
  End If
  n += 1
Loop Until count = 50
 
For i As UInteger = 0 To 9
  Print i; ":" ;
  For j As UInteger = 1 To 5
    Print Using "######"; num(i, j);
  Next j
  Print
Next i
 
Print
Print "Press any key to quit"
Sleep