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


import Stream._
 
object MDR extends App {
 
  def mdr(x: BigInt, base: Int = 10): (BigInt, Long) = {
    def multiplyDigits(x: BigInt): BigInt = ((x.toString(base) map (_.asDigit)) :\ BigInt(1))(_*_)
    def loop(p: BigInt, c: Long): (BigInt, Long) = if (p < base) (p, c) else loop(multiplyDigits(p), c+1)
    loop(multiplyDigits(x), 1)
  }
 
  printf("%15s\t%10s\t%s\n","Number","MDR","MP")
  printf("%15s\t%10s\t%s\n","======","===","==")
  Seq[BigInt](123321, 7739, 893, 899998, BigInt("393900588225"), BigInt("999999999999")) foreach {x =>
    val (s, c) = mdr(x)
    printf("%15s\t%10s\t%2s\n",x,s,c)
  }
  println
 
  val mdrs: Stream[Int] => Stream[(Int, BigInt)] = i => i map (x => (x, mdr(x)._1))
 
  println("MDR: [n0..n4]")
  println("==== ========")
  ((for {i <- 0 to 9} yield (mdrs(from(0)) take 11112 toList) filter {_._2 == i})
    .map {_ take 5} map {xs => xs map {_._1}}).zipWithIndex
    .foreach{p => printf("%3s: [%s]\n",p._2,p._1.mkString(", "))}
 
}