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


 
#include <stdio.h>
 
#define twidth 5
#define mdr(rmdr, rmp, n)\
    do { *rmp = 0; _mdr(rmdr, rmp, n); } while (0)
 
void _mdr(int *rmdr, int *rmp, long long n)
{
    /* Adjust r if 0 case, so we don't return 1 */
    int r = n ? 1 : 0;
    while (n) { 
        r *= (n % 10);
        n /= 10; 
    }
 
    (*rmp)++;
    if (r >= 10)
        _mdr(rmdr, rmp, r);
    else
        *rmdr = r;
}
 
int main(void)
{
    int i, j, vmdr, vmp;
    const int values[] = { 123321, 7739, 893, 899998 };
    const int vsize    = sizeof(values) / sizeof(values[0]);
 
    /* Initial test values */
    printf("Number    MDR    MP\n");
    for (i = 0; i < vsize; ++i) {
        mdr(&vmdr, &vmp, values[i]);
        printf("%6d   %3d   %3d\n", values[i], vmdr, vmp);
    }
 
    /* Determine table values */
    int table[10][twidth] = { 0 };
    int tfill[10]         = { 0 };
    int total             = 0;
    for (i = 0; total < 10 * twidth; ++i) {
        mdr(&vmdr, &vmp, i);
        if (tfill[vmdr] < twidth) {
            table[vmdr][tfill[vmdr]++] = i;
            total++;
        }
    }
 
    /* Print calculated table values */
    printf("\nMDR: [n0..n4]\n");
    for (i = 0; i < 10; ++i) {
        printf("%3d: [", i);
        for (j = 0; j < twidth; ++j)
            printf("%d%s", table[i][j], j != twidth - 1 ? ", " : ""); 
        printf("]\n");
    }
 
    return 0;
}
 