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


 
!Implemented by Anant Dixit (Oct, 2014)
program mdr
implicit none
integer :: i, mdr, mp, n, j
character(len=*), parameter :: hfmt = '(A18)', nfmt = '(I6)'
character(len=*), parameter :: cfmt = '(A3)', rfmt = '(I3)', ffmt = '(I9)'
 
write(*,hfmt) 'Number   MDR   MP '
write(*,*) '------------------'
 
i = 123321
call root_pers(i,mdr,mp)
write(*,nfmt,advance='no') i
write(*,cfmt,advance='no') '   '
write(*,rfmt,advance='no') mdr
write(*,cfmt,advance='no') '   '
write(*,rfmt) mp
 
i = 3939
call root_pers(i,mdr,mp)
write(*,nfmt,advance='no') i
write(*,cfmt,advance='no') '   '
write(*,rfmt,advance='no') mdr
write(*,cfmt,advance='no') '   '
write(*,rfmt) mp
 
i = 8822
call root_pers(i,mdr,mp)
write(*,nfmt,advance='no') i
write(*,cfmt,advance='no') '   '
write(*,rfmt,advance='no') mdr
write(*,cfmt,advance='no') '   '
write(*,rfmt) mp
 
i = 39398
call root_pers(i,mdr,mp)
write(*,nfmt,advance='no') i
write(*,cfmt,advance='no') '   '
write(*,rfmt,advance='no') mdr
write(*,cfmt,advance='no') '   '
write(*,rfmt) mp
 
write(*,*)
write(*,*)
write(*,*) 'First five numbers with MDR in first column: '
write(*,*) '---------------------------------------------'
 
do i = 0,9
  n = 0
  j = 0
  write(*,rfmt,advance='no') i
  do
    call root_pers(j,mdr,mp)
    if(mdr.eq.i) then
      n = n+1
      if(n.eq.5) then
        write(*,ffmt) j
        exit
      else
        write(*,ffmt,advance='no') j
      end if
    end if
    j = j+1
  end do
end do
 
end program
 
subroutine root_pers(i,mdr,mp)
implicit none
integer :: N, s, a, i, mdr, mp
n = i
a = 0
if(n.lt.10) then
  mdr = n
  mp = 0
  return
end if
do while(n.ge.10)
  a = a + 1
  s = 1
  do while(n.gt.0)
    s = s * mod(n,10)
    n = int(real(n)/10.0D0)
  end do
  n = s
end do
mdr = s
mp = a
end subroutine
 
 