Digital root/Multiplicative digital root
(Redirected from Product of decimal digits of n)
You are encouraged to solve this task according to the task description, using any language you may know.
The multiplicative digital root (MDR) and multiplicative persistence (MP) of a number, , is calculated rather like the Digital root except digits are multiplied instead of being added:
- Set to and to .
- While has more than one digit:
- Find a replacement as the multiplication of the digits of the current value of .
- Increment .
- Return (= MP) and (= 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
11l
F mdroot(n)
V count = 0
V mdr = n
L mdr > 9
V m = mdr
V digits_mul = 1
L m != 0
digits_mul *= m % 10
m = m I/ 10
mdr = digits_mul
count++
R (count, mdr)
print(‘Number: (MP, MDR)’)
print(‘====== =========’)
L(n) (123321, 7739, 893, 899998)
print(‘#6: ’.format(n), end' ‘’)
print(mdroot(n))
[[Int]] table
table.resize(10)
V n = 0
L min(table.map(row -> row.len)) < 5
table[mdroot(n)[1]].append(n)
n++
print(‘’)
print(‘MP: [n0..n4]’)
print(‘== ========’)
L(val) table
print(‘#2: ’.format(L.index), end' ‘’)
print(val[0.<5])- Output:
Number: (MP, MDR) ====== ========= 123321: (3, 8) 7739: (3, 8) 893: (3, 2) 899998: (2, 0) MP: [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]
Ada
The solution uses the Package "Generic_Root" from the additive digital roots [[1]].
with Ada.Text_IO, Generic_Root; use Generic_Root;
procedure Multiplicative_Root is
procedure Compute is new Compute_Root("*"); -- "*" for multiplicative roots
package TIO renames Ada.Text_IO;
package NIO is new TIO.Integer_IO(Number);
procedure Print_Numbers(Target_Root: Number; How_Many: Natural) is
Current: Number := 0;
Root, Pers: Number;
begin
for I in 1 .. How_Many loop
loop
Compute(Current, Root, Pers);
exit when Root = Target_Root;
Current := Current + 1;
end loop;
NIO.Put(Current, Width => 6);
if I < How_Many then
TIO.Put(",");
end if;
Current := Current + 1;
end loop;
end Print_Numbers;
Inputs: Number_Array := (123321, 7739, 893, 899998);
Root, Pers: Number;
begin
TIO.Put_Line(" Number MDR MP");
for I in Inputs'Range loop
Compute(Inputs(I), Root, Pers);
NIO.Put(Inputs(I), Width => 8);
NIO.Put(Root, Width => 6);
NIO.Put(Pers, Width => 6);
TIO.New_Line;
end loop;
TIO.New_Line;
TIO.Put_Line(" MDR first_five_numbers_with_that_MDR");
for I in 0 .. 9 loop
TIO.Put(" " & Integer'Image(I) & " ");
Print_Numbers(Target_Root => Number(I), How_Many => 5);
TIO.New_Line;
end loop;
end Multiplicative_Root;
- Output:
Number MDR MP
123321 8 3
7739 8 3
893 2 3
899998 0 2
MDR first_five_numbers_with_that_MDR
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
ALGOL 68
BEGIN # Multiplicative Digital Roots #
# structure to hold the results of calculating the digital root & persistence #
MODE DR = STRUCT( INT root, INT persistence );
# returns the product of the digits of number #
OP DIGITPRODUCT = ( INT number )INT:
BEGIN
INT result := 1;
INT rest := number;
WHILE result TIMESAB ( rest MOD 10 );
rest OVERAB 10;
rest > 0
DO SKIP OD;
result
END; # DIGITPRODUCT #
# calculates the multiplicative digital root and persistence of number #
OP MDROOT = ( INT number )DR:
BEGIN
INT mp := 0;
INT mdr := ABS number;
WHILE mdr > 9 DO
mp +:= 1;
mdr := DIGITPRODUCT mdr
OD;
( mdr, mp )
END; # MDROOT #
# prints a number and its MDR and MP #
PROC print md root = ( INT number )VOID:
BEGIN
DR mdr = MDROOT( number );
print( ( whole( number, -6 ), ": MDR: ", whole( root OF mdr, 0 ), ", MP: ", whole( persistence OF mdr, -2 ), newline ) )
END; # print md root #
# prints the first few numbers with each possible Multiplicative Digital #
# Root. The number of values to print is specified as a parameter #
PROC tabulate mdr = ( INT number of values )VOID:
BEGIN
[ 0 : 9, 1 : number of values ]INT mdr values;
[ 0 : 9 ]INT mdr counts;
mdr counts[ AT 1 ] := ( 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 );
# find the first few numbers with each possible mdr #
INT values found := 0;
INT required values := 10 * number of values;
FOR value FROM 0 WHILE values found < required values DO
DR mdr = MDROOT value;
IF mdr counts[ root OF mdr ] < number of values THEN
# need more values with this multiplicative digital root #
values found +:= 1;
mdr counts[ root OF mdr ] +:= 1;
mdr values[ root OF mdr, mdr counts[ root OF mdr ] ] := value
FI
OD;
# print the values #
print( ( "MDR: [n0..n" + whole( number of values - 1, 0 ) + "]", newline ) );
print( ( "=== ========", newline ) );
FOR mdr pos FROM 1 LWB mdr values TO 1 UPB mdr values DO
STRING separator := ": [";
print( ( whole( mdr pos, -3 ) ) );
FOR val pos FROM 2 LWB mdr values TO 2 UPB mdr values DO
print( ( separator + whole( mdr values[ mdr pos, val pos ], 0 ) ) );
separator := ", "
OD;
print( ( "]", newline ) )
OD
END; # tabulate mdr #
# task test cases #
print md root( 123321 );
print md root( 7739 );
print md root( 893 );
print md root( 899998 );
tabulate mdr( 5 )
END- Output:
123321: MDR: 8, MP: 3 7739: MDR: 8, MP: 3 893: MDR: 2, MP: 3 899998: MDR: 0, MP: 2 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]
ALGOL W
begin
% calculate the Multiplicative Digital Root (mdr) and Multiplicative Persistence (mp) of n %
procedure getMDR ( integer value n
; integer result mdr, mp
) ;
begin
mp := 0;
mdr := abs n;
while mdr > 9 do begin
integer v;
v := mdr;
mdr := 1;
while begin
mdr := mdr * ( v rem 10 );
v := v div 10;
v > 0
end do begin end;
mp := mp + 1;
end while_mdr_gt_9 ;
end getMDR ;
% task test cases %
write( " N MDR MP" );
for n := 123321, 7739, 893, 899998 do begin
integer mdr, mp;
getMDR( n, mdr, mp );
write( s_w := 1, i_w := 8, n, i_w := 3, mdr, i_w := 2, mp )
end for_n ;
begin % find the first 5 numbers with each possible MDR %
integer requiredMdrs;
requiredMdrs := 5;
begin
integer array firstFew ( 0 :: 9, 1 :: requiredMdrs );
integer array mdrFOund ( 0 :: 9 );
integer totalFound, requiredTotal, n;
for i := 0 until 9 do mdrFound( i ) := 0;
totalFound := 0;
requiredTotal := 10 * requiredMdrs;
n := -1;
while totalFound < requiredTotal do begin
integer mdr, mp;
n := n + 1;
getMDR( n, mdr, mp );
if mdrFound( mdr ) < requiredMdrs then begin
% found another number with this MDR and haven't found enough yet %
totalFound := totalFound + 1;
mdrFound( mdr ) := mdrFound( mdr ) + 1;
firstFew( mdr, mdrFound( mdr ) ) := n
end if_found_another_MDR
end while_totalFound_lt_requiredTotal ;
% print the table of MDRs andnumbers %
write( "MDR: [n0..n4]" );
write( "=== ========" );
for v := 0 until 9 do begin
write( i_w := 3, s_w := 0, v, ": [" );
for foundPos := 1 until requiredMdrs do begin
if foundPos > 1 then writeon( s_w := 0, ", " );
writeon( i_w := 1, s_w := 0, firstFew( v, foundPos ) )
end for_foundPos ;
writeon( s_w := 0, "]" )
end for_v
end
end
end.- Output:
N MDR MP
123321 8 3
7739 8 3
893 2 3
899998 0 2
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]
AWK
# Multiplicative Digital Roots
BEGIN {
printMdrAndMp( 123321 );
printMdrAndMp( 7739 );
printMdrAndMp( 893 );
printMdrAndMp( 899998 );
tabulateMdr( 5 );
} # BEGIN
function printMdrAndMp( n )
{
calculateMdrAndMp( n );
printf( "%6d: MDR: %d, MP: %2d\n", n, MDR, MP );
} # printMdrAndMp
function calculateMdrAndMp( n, mdrStr, digit )
{
MP = 0; # global Multiplicative Persistence
MDR = ( n < 0 ? -n : n ); # global Multiplicative Digital Root
while( MDR > 9 )
{
MP ++;
mdrStr = "" MDR;
MDR = 1;
for( digit = 1; digit <= length( mdrStr ); digit ++ )
{
MDR *= ( substr( mdrStr, digit, 1 ) * 1 );
} # for digit
} # while MDR > 9
} # calculateMdrAndMp
function tabulateMdr( n, rqdValues, valueCount, value, pos )
{
# generate a table of the first n numbers with each possible MDR
rqdValues = n * 10;
valueCount = 0;
for( value = 0; valueCount < rqdValues; value ++ )
{
calculateMdrAndMp( value );
if( mdrCount[ MDR ] < n )
{
# still need another value with this MDR
valueCount ++;
mdrCount[ MDR ] ++;
mdrValues[ MDR ":" mdrCount[ MDR ] ] = value;
} # if mdrCount[ MDR ] < n
} # for value
# print the table
printf( "MDR: [n0..n%d]\n", n - 1 );
printf( "=== ========\n" );
for( pos = 0; pos < 10; pos ++ )
{
printf( "%3d:", pos );
separator = " [";
for( value = 1; value <= n; value ++ )
{
printf( "%s%d", separator, mdrValues[ pos ":" value ] );
separator = ", "
} # for value
printf( "]\n" );
} # for pos
} # tabulateMdr
- Output:
123321: MDR: 8, MP: 3 7739: MDR: 8, MP: 3 893: MDR: 2, MP: 3 899998: MDR: 0, MP: 2 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]
Bracmat
(
& ( MP/MDR
= prod L n
. ( prod
= d
. @(!arg:%@?d ?arg)&!d*prod$!arg
| 1
)
& !arg:?L
& whl
' ( @(!arg:? [>1)
& (prod$!arg:?arg) !L:?L
)
& !L:? [?n
& (!n+-1.!arg)
)
& ( test
= n
. !arg:%?n ?arg
& out$(!n "\t:" MP/MDR$!n)
& test$!arg
|
)
& test$(123321 7739 893 899998)
& 0:?i
& 1:?collecting:?done
& whl
' ( !i+1:?i
& MP/MDR$!i:(?MP.?MDR)
& ( !done:?*(!MDR.)^((?.)+?)*?
| (!MDR.)^(!i.)*!collecting:?collecting
& ( !collecting:?A*(!MDR.)^(?is+[5)*?Z
& !A*!Z:?collecting
& (!MDR.)^!is*!done:?done
|
)
)
& !collecting:~1
)
& whl
' ( !done:(?MDR.)^?is*?done
& put$(!MDR ":")
& whl'(!is:(?i.)+?is&put$(!i " "))
& put$\n
)
);- Output:
123321 : (3.8) 7739 : (3.8) 893 : (3.2) 899998 : (2.0) 0 :10 20 25 30 40 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
C
#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;
}
- Output:
Number MDR MP 123321 8 3 7739 8 3 893 2 3 899998 0 2 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]
C#
using System;
using System.Collections.Generic;
using System.Linq;
class Program
{
static Tuple<int, int> DigitalRoot(long num)
{
int mp = 0;
while (num > 9)
{
num = num.ToString().ToCharArray().Select(x => x - '0').Aggregate((a, b) => a * b);
mp++;
}
return new Tuple<int, int>(mp, (int)num);
}
static void Main(string[] args)
{
foreach (long num in new long[] { 123321, 7739, 893, 899998 })
{
var t = DigitalRoot(num);
Console.WriteLine("{0} has multiplicative persistence {1} and multiplicative digital root {2}", num, t.Item1, t.Item2);
}
const int twidth = 5;
List<long>[] table = new List<long>[10];
for (int i = 0; i < 10; i++)
table[i] = new List<long>();
long number = -1;
while (table.Any(x => x.Count < twidth))
{
var t = DigitalRoot(++number);
if (table[t.Item2].Count < twidth)
table[t.Item2].Add(number);
}
for (int i = 0; i < 10; i++)
Console.WriteLine(" {0} : [{1}]", i, string.Join(", ", table[i]));
}
}
- Output:
123321 has multiplicative persistence 3 and multiplicative digital root 8 7739 has multiplicative persistence 3 and multiplicative digital root 8 893 has multiplicative persistence 3 and multiplicative digital root 2 899998 has multiplicative persistence 2 and multiplicative digital root 0 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]
C++
#include <iomanip>
#include <map>
#include <vector>
#include <iostream>
using namespace std;
void calcMDR( int n, int c, int& a, int& b )
{
int m = n % 10; n /= 10;
while( n )
{
m *= ( n % 10 );
n /= 10;
}
if( m >= 10 ) calcMDR( m, ++c, a, b );
else { a = m; b = c; }
}
void table()
{
map<int, vector<int> > mp;
int n = 0, a, b;
bool f = true;
while( f )
{
f = false;
calcMDR( n, 1, a, b );
mp[a].push_back( n );
n++;
for( int x = 0; x < 10; x++ )
if( mp[x].size() < 5 )
{ f = true; break; }
}
cout << "| MDR | [n0..n4]\n+-------+------------------------------------+\n";
for( int x = 0; x < 10; x++ )
{
cout << right << "| " << setw( 6 ) << x << "| ";
for( vector<int>::iterator i = mp[x].begin(); i != mp[x].begin() + 5; i++ )
cout << setw( 6 ) << *i << " ";
cout << "|\n";
}
cout << "+-------+------------------------------------+\n\n";
}
int main( int argc, char* argv[] )
{
cout << "| NUMBER | MDR | MP |\n+----------+----------+----------+\n";
int numbers[] = { 123321, 7739, 893, 899998 }, a, b;
for( int x = 0; x < 4; x++ )
{
cout << right << "| " << setw( 9 ) << numbers[x] << "| ";
calcMDR( numbers[x], 1, a, b );
cout << setw( 9 ) << a << "| " << setw( 9 ) << b << "|\n";
}
cout << "+----------+----------+----------+\n\n";
table();
return system( "pause" );
}
- Output:
| NUMBER | MDR | MP | +----------+----------+----------+ | 123321| 8| 3| | 7739| 8| 3| | 893| 2| 3| | 899998| 0| 2| +----------+----------+----------+ | 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 | +-------+------------------------------------+
CLU
digits = iter (n: int) yields (int)
while n>0 do
yield(n//10)
n := n/10
end
end digits
mdr = proc (n: int) returns (int,int)
i: int := 0
while n>=10 do
m: int := 1
for d: int in digits(n) do
m := m * d
end
n := m
i := i+1
end
return (i,n)
end mdr
first_mdr = iter (target_mdr, n: int) yields (int)
i: int := 0
while n>0 do
x, m: int := mdr(i)
if m=target_mdr then
yield(i)
n := n -1
end
i := i+1
end
end first_mdr
start_up = proc ()
po: stream := stream$primary_output()
nums: sequence[int] := sequence[int]$[123321, 7739, 893, 899998]
stream$putl(po, " N MDR MP")
stream$putl(po, "====== === ==")
for num: int in sequence[int]$elements(nums) do
stream$putright(po, int$unparse(num), 6)
stream$puts(po, " ")
i, m: int := mdr(num)
stream$putright(po, int$unparse(m), 3)
stream$puts(po, " ")
stream$putright(po, int$unparse(i), 3)
stream$putl(po, "")
end
stream$putl(po, "\nMDR: [n0..n4]")
stream$putl(po, "=== ========")
for dgt: int in int$from_to(0,9) do
stream$putright(po, int$unparse(dgt), 3)
stream$puts(po, ": ")
for num: int in first_mdr(dgt, 5) do
stream$puts(po, int$unparse(num) || " ")
end
stream$putl(po, "")
end
end start_up- Output:
N MDR MP ====== === == 123321 8 3 7739 8 3 893 2 3 899998 0 2 MDR: [n0..n4] === ======== 0: 0 10 20 25 30 1: 1 11 111 1111 11111 2: 2 12 21 26 34 3: 3 13 31 113 131 4: 4 14 22 27 39 5: 5 15 35 51 53 6: 6 16 23 28 32 7: 7 17 71 117 171 8: 8 18 24 29 36 9: 9 19 33 91 119
Common Lisp
(defun mdr/p (n)
"Return a list with MDR and MP of n"
(if (< n 10)
(list n 0)
(mdr/p-aux n 1 1)))
(defun mdr/p-aux (n a c)
(cond ((and (zerop n) (< a 10)) (list a c))
((zerop n) (mdr/p-aux a 1 (+ c 1)))
(t (mdr/p-aux (floor n 10) (* (rem n 10) a) c))))
(defun first-n-number-for-each-root (n &optional (r 0) (lst nil) (c 0))
"Return the first m number with MDR = 0 to 9"
(cond ((and (= (length lst) n) (= r 9)) (format t "~3@a: ~a~%" r (reverse lst)))
((= (length lst) n) (format t "~3@a: ~a~%" r (reverse lst))
(first-n-number-for-each-root n (+ r 1) nil 0))
((= (first (mdr/p c)) r) (first-n-number-for-each-root n r (cons c lst) (+ c 1)))
(t (first-n-number-for-each-root n r lst (+ c 1)))))
(defun start ()
(format t "Number: MDR MD~%")
(loop for el in '(123321 7739 893 899998)
do (format t "~6@a: ~{~3@a ~}~%" el (mdr/p el)))
(format t "~%MDR: [n0..n4]~%")
(first-n-number-for-each-root 5))
- Output:
Number: MDR MD 123321: 8 3 7739: 8 3 893: 2 3 899998: 0 2 MDR: [n0..n4] 0: (0 10 20 25 30) 1: (1 11 111 1111 11111) 2: (2 12 21 26 34) 3: (3 13 31 113 131) 4: (4 14 22 27 39) 5: (5 15 35 51 53) 6: (6 16 23 28 32) 7: (7 17 71 117 171) 8: (8 18 24 29 36) 9: (9 19 33 91 119)
Component Pascal
MODULE MDR;
IMPORT StdLog, Strings, TextMappers, DevCommanders;
PROCEDURE CalcMDR(x: LONGINT; OUT mdr, mp: LONGINT);
VAR
str: ARRAY 64 OF CHAR;
i: INTEGER;
BEGIN
mdr := 1; mp := 0;
LOOP
Strings.IntToString(x,str);
IF LEN(str$) = 1 THEN mdr := x; EXIT END;
i := 0;mdr := 1;
WHILE i < LEN(str$) DO
mdr := mdr * (ORD(str[i]) - ORD('0'));
INC(i)
END;
INC(mp);
x := mdr
END
END CalcMDR;
PROCEDURE Do*;
VAR
mdr,mp: LONGINT;
s: TextMappers.Scanner;
BEGIN
s.ConnectTo(DevCommanders.par.text);
s.SetPos(DevCommanders.par.beg);
REPEAT
s.Scan;
IF (s.type = TextMappers.int) OR (s.type = TextMappers.lint) THEN
CalcMDR(s.int,mdr,mp);
StdLog.Int(s.int);
StdLog.String(" MDR: ");StdLog.Int(mdr);
StdLog.String(" MP: ");StdLog.Int(mp);StdLog.Ln
END
UNTIL s.rider.eot;
END Do;
PROCEDURE Show(i: INTEGER; x: ARRAY OF LONGINT);
VAR
k: INTEGER;
BEGIN
StdLog.Int(i);StdLog.String(": ");
FOR k := 0 TO LEN(x) - 1 DO
StdLog.Int(x[k])
END;
StdLog.Ln
END Show;
PROCEDURE FirstFive*;
VAR
i,j: INTEGER;
five: ARRAY 5 OF LONGINT;
x,mdr,mp: LONGINT;
BEGIN
FOR i := 0 TO 9 DO
j := 0;x := 0;
WHILE (j < LEN(five)) DO
CalcMDR(x,mdr,mp);
IF mdr = i THEN five[j] := x; INC(j) END;
INC(x)
END;
Show(i,five)
END
END FirstFive;
END MDR.
Execute: ^Q MDR.Do 123321 7739 893 899998 ~
- Output:
123321 MDR: 8 MP: 3 7739 MDR: 8 MP: 3 893 MDR: 2 MP: 3 899998 MDR: 0 MP: 2
Execute: ^Q MDR.FirstFive
- Output:
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
D
import std.stdio, std.algorithm, std.typecons, std.range, std.conv;
/// Multiplicative digital root.
auto mdRoot(in int n) pure /*nothrow*/ {
auto mdr = [n];
while (mdr.back > 9)
mdr ~= reduce!q{a * b}(1, mdr.back.text.map!(d => d - '0'));
//mdr ~= mdr.back.text.map!(d => d - '0').mul;
//mdr ~= mdr.back.reverseDigits.mul;
return tuple(mdr.length - 1, mdr.back);
}
void main() {
"Number: (MP, MDR)\n====== =========".writeln;
foreach (immutable n; [123321, 7739, 893, 899998])
writefln("%6d: (%s, %s)", n, n.mdRoot[]);
auto table = (int[]).init.repeat.enumerate!int.take(10).assocArray;
auto n = 0;
while (table.byValue.map!walkLength.reduce!min < 5) {
table[n.mdRoot[1]] ~= n;
n++;
}
"\nMP: [n0..n4]\n== ========".writeln;
foreach (const mp; table.byKey.array.sort())
writefln("%2d: %s", mp, table[mp].take(5));
}
- Output:
Number: (MP, MDR) ====== ========= 123321: (3, 8) 7739: (3, 8) 893: (3, 2) 899998: (2, 0) MP: [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]
Alternative Version
import std.stdio, std.algorithm, std.typecons, std.range;
uint digitsProduct(uint n) pure nothrow @nogc {
typeof(return) result = !!n;
while (n) {
result *= n % 10;
n /= 10;
}
return result;
}
/// Multiplicative digital root.
Tuple!(size_t, uint) mdRoot(uint m) pure nothrow {
auto mdr = m
.recurrence!((a, n) => a[n - 1].digitsProduct)
.until!q{ a <= 9 }(OpenRight.no).array;
return tuple(mdr.length - 1, mdr.back);
}
void main() {
"Number: (MP, MDR)\n====== =========".writeln;
foreach (immutable n; [123321, 7739, 893, 899998])
writefln("%6d: (%s, %s)", n, n.mdRoot[]);
auto table = (int[]).init.repeat.enumerate!int.take(10).assocArray;
auto n = 0;
while (table.byValue.map!walkLength.reduce!min < 5) {
table[n.mdRoot[1]] ~= n;
n++;
}
"\nMP: [n0..n4]\n== ========".writeln;
foreach (const mp; table.byKey.array.sort())
writefln("%2d: %s", mp, table[mp].take(5));
}
More Efficient Version
import std.stdio, std.algorithm, std.range;
/// Multiplicative digital root.
uint[2] mdRoot(in uint n) pure nothrow @nogc {
uint mdr = n;
uint count = 0;
while (mdr > 9) {
uint m = mdr;
uint digitsMul = !!m;
while (m) {
digitsMul *= m % 10;
m /= 10;
}
mdr = digitsMul;
count++;
}
return [count, mdr];
}
void main() {
"Number: [MP, MDR]\n====== =========".writeln;
foreach (immutable n; [123321, 7739, 893, 899998])
writefln("%6d: %s", n, n.mdRoot);
auto table = (int[]).init.repeat.enumerate!int.take(10).assocArray;
auto n = 0;
while (table.byValue.map!walkLength.reduce!min < 5) {
table[n.mdRoot[1]] ~= n;
n++;
}
"\nMP: [n0..n4]\n== ========".writeln;
foreach (const mp; table.byKey.array.sort())
writefln("%2d: %s", mp, table[mp].take(5));
}
The output is similar.
EasyLang
proc _mdr n . md mp .
if n > 0
r = 1
.
while n > 0
r *= n mod 10
n = n div 10
.
mp += 1
if r >= 10
_mdr r md mp
else
md = r
.
.
proc mdr n . md mp .
mp = 0
_mdr n md mp
.
numfmt 0 6
print "Number MDR MP"
for v in [ 123321 7739 893 899998 ]
mdr v md mp
print v & md & mp
.
width = 5
len table[] 10 * width
arrbase table[] 0
len tfill[] 10
arrbase tfill[] 0
numfmt 0 0
while total < 10 * width
mdr i md mp
if tfill[md] < width
table[md * width + tfill[md]] = i
tfill[md] += 1
total += 1
.
i += 1
.
print "\nMDR: [n0..n4]"
for i = 0 to 9
write i & ": ["
for j = 0 to width - 1
write table[i * width + j]
if j < width - 1
write ","
.
.
print "]"
.
- Output:
Number MDR MP 123321 8 3 7739 8 3 893 2 3 899998 0 2 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]
Elixir
defmodule Digital do
def mdroot(n), do: mdroot(n, 0)
defp mdroot(n, persist) when n < 10, do: {n, persist}
defp mdroot(n, persist), do: mdroot(product(n, 1), persist+1)
defp product(0, prod), do: prod
defp product(n, prod), do: product(div(n, 10), prod*rem(n, 10))
def task1(data) do
IO.puts "Number: MDR MP\n====== === =="
Enum.each(data, fn n ->
{mdr, persist} = mdroot(n)
:io.format "~6w: ~w ~2w~n", [n, mdr, persist]
end)
end
def task2(m \\ 5) do
IO.puts "\nMDR: [n0..n#{m-1}]\n=== ========"
map = add_map(0, m, Map.new)
Enum.each(0..9, fn i ->
first = map[i] |> Enum.reverse |> Enum.take(m)
IO.puts " #{i}: #{inspect first}"
end)
end
defp add_map(n, m, map) do
{mdr, _persist} = mdroot(n)
new_map = Map.update(map, mdr, [n], fn vals -> [n | vals] end)
min_len = Map.values(new_map) |> Enum.map(&length(&1)) |> Enum.min
if min_len < m, do: add_map(n+1, m, new_map),
else: new_map
end
end
Digital.task1([123321, 7739, 893, 899998])
Digital.task2
- Output:
Number: MDR MP ====== === == 123321: 8 3 7739: 8 3 893: 2 3 899998: 0 2 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]
F#
// mdr. Nigel Galloway: June 29th., 2021
let rec fG n g=if n=0 then g else fG(n/10)(g*(n%10))
let mdr n=let rec mdr n g=if n<10 then (n,g) else mdr(fG n 1)(g+1) in mdr n 0
[123321; 7739; 893; 899998] |> List.iter(fun i->let n,g=mdr i in printfn "%d has mdr=%d with persitance %d" i n g)
let fN g=Seq.initInfinite id|>Seq.filter((mdr>>fst>>(=)g))|>Seq.take 5
seq{0..9}|>Seq.iter(fun n->printf "First 5 numbers with mdr %d -> " n; Seq.initInfinite id|>Seq.filter((mdr>>fst>>(=)n))|>Seq.take 5|>Seq.iter(printf "%d ");printfn "")
- Output:
123321 has mdr=8 with persitance 3 7739 has mdr=8 with persitance 3 893 has mdr=2 with persitance 3 899998 has mdr=0 with persitance 2 First 5 numbers with mdr 0 -> 0 10 20 25 30 First 5 numbers with mdr 1 -> 1 11 111 1111 11111 First 5 numbers with mdr 2 -> 2 12 21 26 34 First 5 numbers with mdr 3 -> 3 13 31 113 131 First 5 numbers with mdr 4 -> 4 14 22 27 39 First 5 numbers with mdr 5 -> 5 15 35 51 53 First 5 numbers with mdr 6 -> 6 16 23 28 32 First 5 numbers with mdr 7 -> 7 17 71 117 171 First 5 numbers with mdr 8 -> 8 18 24 29 36 First 5 numbers with mdr 9 -> 9 19 33 91 119
Factor
USING: arrays formatting fry io kernel lists lists.lazy math
math.text.utils prettyprint sequences ;
IN: rosetta-code.multiplicative-digital-root
: mdr ( n -- {persistence,root} )
0 swap
[ 1 digit-groups dup length 1 > ] [ product [ 1 + ] dip ] while
dup empty? [ drop { 0 } ] when first 2array ;
: print-mdr ( n -- )
dup [ 1array ] dip mdr append
"%-12d has multiplicative persistence %d and MDR %d.\n"
vprintf ;
: first5 ( n -- seq ) ! first 5 numbers with MDR of n
0 lfrom swap '[ mdr second _ = ] lfilter 5 swap ltake list>array ;
: print-first5 ( i n -- )
"%-5d" printf bl first5 [ "%-5d " printf ] each nl ;
: header ( -- )
"MDR | First five numbers with that MDR" print
"--------------------------------------" print ;
: first5-table ( -- )
header 10 iota [ print-first5 ] each-index ;
: main ( -- )
{ 123321 7739 893 899998 } [ print-mdr ] each nl first5-table ;
MAIN: main
- Output:
123321 has multiplicative persistence 3 and MDR 8. 7739 has multiplicative persistence 3 and MDR 8. 893 has multiplicative persistence 3 and MDR 2. 899998 has multiplicative persistence 2 and MDR 0. MDR | First five numbers with that MDR -------------------------------------- 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
Fortran
!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
Number MDR MP ------------------ 123321 8 3 3939 2 4 8822 0 3 39398 0 3 First five numbers with MDR in first column: --------------------------------------------- 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
FreeBASIC
' 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
- Output:
123321 MDR = 8 MP = 3 7739 MDR = 8 MP = 3 893 MDR = 2 MP = 3 899998 MDR = 0 MP = 2 MDR 1 2 3 4 5 === =========================== 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
Go
package main
import "fmt"
// Only valid for n > 0 && base >= 2
func mult(n uint64, base int) (mult uint64) {
for mult = 1; mult > 0 && n > 0; n /= uint64(base) {
mult *= n % uint64(base)
}
return
}
// Only valid for n >= 0 && base >= 2
func MultDigitalRoot(n uint64, base int) (mp, mdr int) {
var m uint64
for m = n; m >= uint64(base); mp++ {
m = mult(m, base)
}
return mp, int(m)
}
func main() {
const base = 10
const size = 5
const testFmt = "%20v %3v %3v\n"
fmt.Printf(testFmt, "Number", "MDR", "MP")
for _, n := range [...]uint64{
123321, 7739, 893, 899998,
18446743999999999999,
// From http://mathworld.wolfram.com/MultiplicativePersistence.html
3778888999, 277777788888899,
} {
mp, mdr := MultDigitalRoot(n, base)
fmt.Printf(testFmt, n, mdr, mp)
}
fmt.Println()
var list [base][]uint64
for i := range list {
list[i] = make([]uint64, 0, size)
}
for cnt, n := size*base, uint64(0); cnt > 0; n++ {
_, mdr := MultDigitalRoot(n, base)
if len(list[mdr]) < size {
list[mdr] = append(list[mdr], n)
cnt--
}
}
const tableFmt = "%3v: %v\n"
fmt.Printf(tableFmt, "MDR", "First")
for i, l := range list {
fmt.Printf(tableFmt, i, l)
}
}
- Output:
Number MDR MP
123321 8 3
7739 8 3
893 2 3
899998 0 2
18446743999999999999 0 2
3778888999 0 10
277777788888899 0 11
MDR: First
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]
Haskell
Note that in the function mdrNums we don't know in advance how many numbers we'll need to examine to find the first 5 associated with all the MDRs. Using a lazy array to accumulate these numbers allows us to keep the function simple.
import Control.Arrow
import Data.Array
import Data.LazyArray
import Data.List (unfoldr)
import Data.Tuple
import Text.Printf
-- The multiplicative persistence (MP) and multiplicative digital root (MDR) of
-- the argument.
mpmdr :: Integer -> (Int, Integer)
mpmdr = (length *** head) . span (> 9) . iterate (product . digits)
-- Pairs (mdr, ns) where mdr is a multiplicative digital root and ns are the
-- first k numbers having that root.
mdrNums :: Int -> [(Integer, [Integer])]
mdrNums k = assocs $ lArrayMap (take k) (0,9) [(snd $ mpmdr n, n) | n <- [0..]]
digits :: Integral t => t -> [t]
digits 0 = [0]
digits n = unfoldr step n
where step 0 = Nothing
step k = Just (swap $ quotRem k 10)
printMpMdrs :: [Integer] -> IO ()
printMpMdrs ns = do
putStrLn "Number MP MDR"
putStrLn "====== == ==="
sequence_ [printf "%6d %2d %2d\n" n p r | n <- ns, let (p,r) = mpmdr n]
printMdrNums:: Int -> IO ()
printMdrNums k = do
putStrLn "MDR Numbers"
putStrLn "=== ======="
let showNums = unwords . map show
sequence_ [printf "%2d %s\n" mdr $ showNums ns | (mdr,ns) <- mdrNums k]
main :: IO ()
main = do
printMpMdrs [123321, 7739, 893, 899998]
putStrLn ""
printMdrNums 5
- Output:
Note that the values in the first column of the table are MDRs, as shown in the task's sample output, not MP as incorrectly stated in the task statement and column header.
Number MP MDR ====== == === 123321 3 8 7739 3 8 893 3 2 899998 2 0 MDR Numbers === ======= 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
Icon and Unicon
Works in both languages:
procedure main(A)
write(right("n",8)," ",right("MP",8),right("MDR",5))
every r := mdr(n := 123321|7739|893|899998) do
write(right(n,8),":",right(r[1],8),right(r[2],5))
write()
write(right("MDR",5)," ","[n0..n4]")
every m := 0 to 9 do {
writes(right(m,5),": [")
every writes(right((m = mdr(n := seq(m))[2],.n)\5,6))
write("]")
}
end
procedure mdr(m)
i := 0
while (.m > 10, m := multd(m), i+:=1)
return [i,m]
end
procedure multd(m)
c := 1
while m > 0 do c *:= 1(m%10, m/:=10)
return c
end
- Output:
->drmdr
n MP MDR
123321: 3 8
7739: 3 8
893: 3 2
899998: 2 0
MDR [n0..n4]
0: [ 0 20 30 40 45]
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]
->
J
First, we need something to split a number into digits:
10&#.inv 123321
1 2 3 3 2 1
Second, we need to find their product:
*/@(10&#.inv) 123321
36
Then we use this inductively until it converges:
*/@(10&#.inv)^:a: 123321
123321 36 18 8
MP is one less than the length of this list, and MDR is the last element of this list:
(<:@#,{:) */@(10&#.inv)^:a: 123321
3 8
(<:@#,{:) */@(10&#.inv)^:a: 7739
3 8
(<:@#,{:) */@(10&#.inv)^:a: 893
3 2
(<:@#,{:) */@(10&#.inv)^:a: 899998
2 0
For the table, we don't need that whole list, we only need the final value. Then use these values to classify the original argument (taking the first five from each group):
(5&{./.~ (*/@(10&#.inv)^:_)"0) i.20000
0 10 20 25 30
1 11 111 1111 11111
2 12 21 26 34
3 13 31 113 131
4 14 22 27 39
5 15 35 51 53
6 16 23 28 32
7 17 71 117 171
8 18 24 29 36
9 19 33 91 119
Note that since the first 10 non-negative integers are single digit values, the first column here doubles as a label (representing the corresponding multiplicative digital root).
Java
import java.util.*;
public class MultiplicativeDigitalRoot {
public static void main(String[] args) {
System.out.println("NUMBER MDR MP");
for (long n : new long[]{123321, 7739, 893, 899998}) {
long[] a = multiplicativeDigitalRoot(n);
System.out.printf("%6d %4d %4d%n", a[0], a[1], a[2]);
}
System.out.println();
Map<Long, List<Long>> table = new HashMap<>();
for (long i = 0; i < 10; i++)
table.put(i, new ArrayList<>());
for (long cnt = 0, n = 0; cnt < 10;) {
long[] res = multiplicativeDigitalRoot(n++);
List<Long> list = table.get(res[1]);
if (list.size() < 5) {
list.add(res[0]);
cnt = list.size() == 5 ? cnt + 1 : cnt;
}
}
System.out.println("MDR: first five numbers with same MDR");
table.forEach((key, lst) -> {
System.out.printf("%3d: ", key);
lst.forEach(e -> System.out.printf("%6s ", e));
System.out.println();
});
}
public static long[] multiplicativeDigitalRoot(long n) {
int mp = 0;
long mdr = n;
while (mdr > 9) {
long m = mdr;
long total = 1;
while (m > 0) {
total *= m % 10;
m /= 10;
}
mdr = total;
mp++;
}
return new long[]{n, mdr, mp};
}
}
NUMBER MDR MP 123321 8 3 7739 8 3 893 2 3 899998 0 2 MDR: first five numbers with same MDR 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
jq
def do_until(condition; next):
def u: if condition then . else (next|u) end;
u;
def mdroot(n):
def multiply: reduce .[] as $i (1; .*$i);
# state: [mdr, persist]
[n, 0]
| do_until( .[0] < 10;
[(.[0] | tostring | explode | map(.-48) | multiply), .[1] + 1]
);
# Produce a table with 10 rows (numbered from 0),
# showing the first n numbers having the row-number as the mdr
def tabulate(n):
# state: [answer_matrix, next_i]
def tab:
def minlength: map(length) | min;
.[0] as $matrix
| .[1] as $i
| if (.[0]|minlength) == n then .[0]
else (mdroot($i) | .[0]) as $mdr
| if $matrix[$mdr]|length < n then
($matrix[$mdr] + [$i]) as $row
| $matrix | setpath([$mdr]; $row)
else $matrix
end
| [ ., $i + 1 ]
| tab
end;
[[], 0] | tab;Example:
def neatly:
. as $in
| range(0;length)
| "\(.): \($in[.])";
def rjust(n): tostring | (n-length)*" " + .;
# The task:
" i : [MDR, MP]",
((123321, 7739, 893, 899998) as $i
| "\($i|rjust(6)): \(mdroot($i))"),
"",
"Tabulation",
"MDR: [n0..n4]",
(tabulate(5) | neatly)- Output:
$ jq -n -r -c -f mdr.jq
i : [MDR, MP]
123321: [8,3]
7739: [8,3]
893: [2,3]
899998: [0,2]
Tabulation
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]
Julia
Function
function digitalmultroot{S<:Integer,T<:Integer}(n::S, bs::T=10)
-1 < n && 1 < bs || throw(DomainError())
ds = n
pers = 0
while bs <= ds
ds = prod(digits(ds, bs))
pers += 1
end
return (pers, ds)
end
Main
const bs = 10
const excnt = 5
println("Testing Multiplicative Digital Root.\n")
for i in [123321, 7739, 893, 899998]
(pers, ds) = digitalmultroot(i, bs)
print(@sprintf("%8d", i))
print(" has persistence ", pers)
println(" and digital root ", ds)
end
dmr = zeros(Int, bs, excnt)
hasroom = trues(bs)
dex = ones(Int, bs)
i = 0
while any(hasroom)
(pers, ds) = digitalmultroot(i, bs)
ds += 1
if hasroom[ds]
dmr[ds, dex[ds]] = i
dex[ds] += 1
if dex[ds] > excnt
hasroom[ds] = false
end
end
i += 1
end
println("\n MDR: First ", excnt, " numbers having this MDR")
for (i, d) in enumerate(0:(bs-1))
print(@sprintf("%4d: ", d))
println(join([@sprintf("%6d", dmr[i, j]) for j in 1:excnt], ","))
end
- Output:
Testing Multiplicative Digital Root.
123321 has persistence 3 and digital root 8
7739 has persistence 3 and digital root 8
893 has persistence 3 and digital root 2
899998 has persistence 2 and digital root 0
MDR: First 5 numbers having this MDR
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
Kotlin
// version 1.1.2
fun multDigitalRoot(n: Int): Pair<Int, Int> = when {
n < 0 -> throw IllegalArgumentException("Negative numbers not allowed")
else -> {
var mdr: Int
var mp = 0
var nn = n
do {
mdr = if (nn > 0) 1 else 0
while (nn > 0) {
mdr *= nn % 10
nn /= 10
}
mp++
nn = mdr
}
while (mdr >= 10)
Pair(mdr, mp)
}
}
fun main(args: Array<String>) {
val ia = intArrayOf(123321, 7739, 893, 899998)
for (i in ia) {
val (mdr, mp) = multDigitalRoot(i)
println("${i.toString().padEnd(9)} MDR = $mdr MP = $mp")
}
println()
println("MDR n0 n1 n2 n3 n4")
println("=== ===========================")
val ia2 = Array(10) { IntArray(6) } // all zero by default
var n = 0
var count = 0
do {
val (mdr, _) = multDigitalRoot(n)
if (ia2[mdr][0] < 5) {
ia2[mdr][0]++
ia2[mdr][ia2[mdr][0]] = n
count++
}
n++
}
while (count < 50)
for (i in 0..9) {
print("$i:")
for (j in 1..5) print("%6d".format(ia2[i][j]))
println()
}
}
- Output:
123321 MDR = 8 MP = 3 7739 MDR = 8 MP = 3 893 MDR = 2 MP = 3 899998 MDR = 0 MP = 2 MDR n0 n1 n2 n3 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
M2000 Interpreter
multDigitalRoot=lambda (n as decimal) ->{
if n<0 then error "Negative numbers not allowed"
def decimal mdr, mp, nn
nn=n
do
mdr=IF(nn>0->1@, 0@)
while nn>0
mdr*=nn mod 10@
nn|div 10@
end while
mp++
nn=mdr
when mdr>=10
=(mdr, mp)
}
Document doc$
ia=(123321, 7739, 893, 899998)
in_ia=each(ia)
while in_ia
(mdr, mp)=multDigitalRoot(array(in_ia))
doc$=format$("{0::-9} mdr = {1} MP = {2}", array(in_ia), mdr, mp)+{
}
end while
let n=0@, count=0&
dim ia2(0 to 9, 0 to 5)
do
mdr=multDigitalRoot(n)#val(0)
if ia2(mdr, 0)<5 then
ia2(mdr, 0)++
ia2(mdr, ia2(mdr, 0))=n
count++
end if
n++
when count<50
doc$={MDR n0 n1 n2 n3 n4
}
doc$={=== ============================
}
for i=0 to 9
doc$=format$("{0}: ", i)
for j=1 to 5
doc$=format$("{0::-6}", ia2(i, j))
next
doc$={
}
next
Clipboard doc$
// Print like in a file (-2 is for console):
Print #-2, doc$- Output:
123321 mdr = 8 MP = 3
7739 mdr = 8 MP = 3
893 mdr = 2 MP = 3
899998 mdr = 0 MP = 2
MDR n0 n1 n2 n3 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
Mathematica / Wolfram Language
ClearAll[mdr, mp, nums];
mdr[n_] := NestWhile[Times @@ IntegerDigits[#] &, n, # > 9 &];
mp[n_] := Length@NestWhileList[Times @@ IntegerDigits[#] &, n, # > 9 &] - 1;
TableForm[{#, mdr[#], mp[#]} & /@ {123321, 7739, 893, 899998},
TableHeadings -> {None, {"Number", "MDR", "MP"}}]
nums = ConstantArray[{}, 10];
For[i = 0, Min[Length /@ nums] < 5, i++, AppendTo[nums[[mdr[i] + 1]], i]];
TableForm[Table[{i, Take[nums[[i + 1]], 5]}, {i, 0, 9}],
TableHeadings -> {None, {"MDR", "First 5"}}, TableDepth -> 2]
- Output:
Number MDR MP
-----------------
123321 8 3
7739 8 3
893 2 3
899998 0 2
MDR First 5
-----------------------------
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}
Nim
import strutils, sequtils, sugar
proc mdroot(n: int): tuple[mp, mdr: int] =
var mdr = @[n]
while mdr[mdr.high] > 9:
var n = 1
for dig in $mdr[mdr.high]:
n *= parseInt($dig)
mdr.add n
(mdr.high, mdr[mdr.high])
for n in [123321, 7739, 893, 899998]:
echo align($n, 6)," ",mdroot(n)
echo ""
var table = newSeqWith(10, newSeq[int]())
for n in 0..int.high:
if table.map((x: seq[int]) => x.len).min >= 5: break
table[mdroot(n).mdr].add n
for mp, val in table:
echo mp, ": ", val[0..4]
- Output:
123321 (mp: 3, mdr: 8) 7739 (mp: 3, mdr: 8) 893 (mp: 3, mdr: 2) 899998 (mp: 2, mdr: 0) 0: @[0, 10, 20, 25, 30] 1: @[1, 11, 111, 1111, 11111] 2: @[2, 12, 21, 26, 34] 3: @[3, 13, 31, 113, 131] 4: @[4, 14, 22, 27, 39] 5: @[5, 15, 35, 51, 53] 6: @[6, 16, 23, 28, 32] 7: @[7, 17, 71, 117, 171] 8: @[8, 18, 24, 29, 36] 9: @[9, 19, 33, 91, 119]
PARI/GP
a(n)=my(i);while(n>9,n=factorback(digits(n));i++);[i,n];
apply(a, [123321, 7739, 893, 899998])
v=vector(10,i,[]); forstep(n=0,oo,1, t=a(n)[2]+1; if(#v[t]<5,v[t]=concat(v[t],n); if(vecmin(apply(length,v))>4, return(v))))- Output:
%1 = [[3, 8], [3, 8], [3, 2], [2, 0]] %2 = [[0, 10, 20, 25, 30], [1, 11, 111, 1111, 11111], [2, 12, 21, 26, 34], [3, 13, 31, 113, 131], [4, 14, 22, 27, 39], [5, 15, 35, 51, 53], [6, 16, 23, 28, 32], [7, 17, 71, 117, 171], [8, 18, 24, 29, 36], [9, 19, 33, 91, 119]]
Pascal
Free Pascal
inspired by Worthwhile_task_shaving :-)
Brute force speed up GetMulDigits.
program MultRoot;
{$IFDEF FPC}
{$MODE DELPHI}{$OPTIMIZATION ON,ALL}{$CODEALIGN proc=16}
{$ENDIF}
{$IFDEF WINDOWS}
{$APPTYPE CONSOLE}
{$ENDIF}
uses
sysutils;
type
tMul3Dgt = array[0..999] of Uint32;
tMulRoot = record
mrNum,
mrMul,
mrPers : Uint64;
end;
const
Testnumbers : array[0..16] of Uint64 =(123321,7739,893,899998,
18446743999999999999,
//first occurence of persistence 0..11
0,10,25,39,77,679, 6788, 68889, 2677889,
26888999, 3778888999, 277777788888899);
var
Mul3Dgt : tMul3Dgt;
procedure InitMulDgt;
var
i,j,k,l : Int32;
begin
l := 999;
For i := 9 downto 0 do
For j := 9 downto 0 do
For k := 9 downto 0 do
Begin
Mul3Dgt[l] := i*j*k;
dec(l);
end;
end;
function GetMulDigits(n:Uint64):UInt64;inline;
var
pMul3Dgt :^tMul3Dgt;
q :Uint64;
begin
pMul3Dgt := @Mul3Dgt[0];
result := 1;
while n >= 1000 do
begin
q := n div 1000;
result *= pMul3Dgt^[n-1000*q];
n := q;
end;
If n>=100 then
result *= pMul3Dgt^[n]
else
if n>=10 then
result *= pMul3Dgt^[n+100]
else
result *= n;//Mul3Dgt[n+110]
end;
procedure GetMulRoot(var MulRoot:tMulRoot);
var
mr,
pers : UInt64;
Begin
pers := 0;
mr := MulRoot.mrNum;
while mr >=10 do
Begin
mr := GetMulDigits(mr);
inc(pers);
end;
MulRoot.mrMul:= mr;
MulRoot.mrPers:= pers;
end;
const
MaxDgtCount = 9;
var
//all initiated with 0
MulRoot:tMulRoot;
Sol : array[0..9,0..MaxDgtCount-1] of tMulRoot;
SolIds : array[0..9] of Int32;
i,idx,mr,AlreadyDone : Int32;
BEGIN
InitMulDgt;
AlreadyDone := 10;//0..9
MulRoot.mrNum := 0;
repeat
GetMulRoot(MulRoot);
mr := MulRoot.mrMul;
idx := SolIds[mr];
If idx<MaxDgtCount then
begin
Sol[mr,idx]:= MulRoot;
inc(idx);
SolIds[mr]:= idx;
if idx =MaxDgtCount then
dec(AlreadyDone);
end;
inc(MulRoot.mrNum);
until AlreadyDone = 0;
writeln('MDR: First');
For i := 0 to 9 do
begin
write(i:3,':');
For idx := 0 to MaxDgtCount-1 do
write(Sol[i,idx].mrNum:MaxDgtCount+1);
writeln;
end;
writeln;
writeln('number':20,' mulroot persitance');
For i := 0 to High(Testnumbers) do
begin
MulRoot.mrNum := Testnumbers[i];
GetMulRoot(MulRoot);
With MulRoot do
writeln(mrNum:20,mrMul:8,mrPers:8);
end;
{$IFDEF WINDOWS}
readln;
{$ENDIF}
END.
- @TIO.RUN:
Real time: 1.580 s CPU share: 99.59 % inline GetMulDigits ->runtime 100%->76%
MDR: First
0: 0 10 20 25 30 40 45 50 52
1: 1 11 111 1111 11111 111111 1111111 11111111 111111111
2: 2 12 21 26 34 37 43 62 73
3: 3 13 31 113 131 311 1113 1131 1311
4: 4 14 22 27 39 41 72 89 93
5: 5 15 35 51 53 57 75 115 135
6: 6 16 23 28 32 44 47 48 61
7: 7 17 71 117 171 711 1117 1171 1711
8: 8 18 24 29 36 38 42 46 49
9: 9 19 33 91 119 133 191 313 331
number mulroot persistence
123321 8 3
7739 8 3
893 2 3
899998 0 2
18446743999999999999 0 2
0 0 0
10 0 1
25 0 2
39 4 3
77 8 4
679 6 5
6788 0 6
68889 0 7
2677889 0 8
26888999 0 9
3778888999 0 10
277777788888899 0 11
PascalABC.NET
##
function MulDigitalRoot(num: int64): (integer, integer);
begin
var mulpersistence := 0;
while num > 9 do
begin
num := num.ToString.ToCharArray.Select(x -> x.ToDigit).Aggregate((x, y) -> x * y);
mulpersistence += 1;
end;
result := (mulpersistence, integer(num));
end;
var nums := |123321, 7739, 893, 899998|;
foreach var num in nums do
begin
var t := MulDigitalRoot(num);
Writeln(num, ' has multiplicative persistence ', t[0], ' and multiplicative digital root ', t[1])
end;
var twidth := 5;
var table := new List<int64>[10];
for var i := 0 to 9 do
table[i] := new List<int64>();
var number := -1;
while (table.Any(x -> x.Count < twidth)) do
begin
number += 1;
var t := MulDigitalRoot(number);
if (table[t[1]].Count < twidth) then
table[t[1]].Add(number);
end;
for var i := 0 to 9 do
WriteLn(i, ' : [', string.Join(', ', table[i]), ']');
- Output:
123321 has multiplicative persistence 3 and multiplicative digital root 8 7739 has multiplicative persistence 3 and multiplicative digital root 8 893 has multiplicative persistence 3 and multiplicative digital root 2 899998 has multiplicative persistence 2 and multiplicative digital root 0 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]
Perl
use warnings;
use strict;
sub mdr {
my $n = shift;
my($count, $mdr) = (0, $n);
while ($mdr > 9) {
my($m, $dm) = ($mdr, 1);
while ($m) {
$dm *= $m % 10;
$m = int($m/10);
}
$mdr = $dm;
$count++;
}
($count, $mdr);
}
print "Number: (MP, MDR)\n====== =========\n";
foreach my $n (123321, 7739, 893, 899998) {
printf "%6d: (%d, %d)\n", $n, mdr($n);
}
print "\nMP: [n0..n4]\n== ========\n";
foreach my $target (0..9) {
my $i = 0;
my @n = map { $i++ while (mdr($i))[1] != $target; $i++; } 1..5;
print " $target: [", join(", ", @n), "]\n";
}
- Output:
Number: (MP, MDR) ====== ========= 123321: (3, 8) 7739: (3, 8) 893: (3, 2) 899998: (2, 0) MP: [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]
PicoLisp
(de mdr-mp (N)
"Returns the solutions in a list, i.e., '(MDR MP)"
(let MP 0
(while (< 1 (length N))
(setq N (apply * (mapcar format (chop N))))
(inc 'MP) )
(list N MP) ) )
# Get the MDR/MP of these nums.
(setq Test-nums '(123321 7739 893 899998))
(let Fmt (6 5 5)
(tab Fmt "Values" "MDR" "MP")
(tab Fmt "======" "===" "==")
(for I Test-nums
(let MDR-MP (mdr-mp I)
(tab Fmt I (car MDR-MP) (cadr MDR-MP)) ) ) )
(prinl)
# Get the nums of these MDRs.
(setq *Want 5)
(setq *Solutions (make (for MDR (range 0 9)
(link (make (let N 0 (until (= *Want (length (made)))
(when (= MDR (car (mdr-mp N)))
(link N) )
(inc 'N) )))) )))
(let Fmt (3 1 -1)
(tab Fmt "MDR" ": " "Values")
(tab Fmt "===" " " "======")
(for (I . S) *Solutions
(tab Fmt (dec I) ": " (glue ", " S)) ) )- Output:
Values MDR MP ====== === == 123321 8 3 7739 8 3 893 2 3 899998 0 2 MDR: Values === ====== 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
Phix
with javascript_semantics function mdr_mp(integer m) integer mp = 0 while m>9 do integer newm = 1 while m do newm *= remainder(m,10) m = floor(m/10) end while m = newm mp += 1 end while return {m,mp} end function constant tests = {123321, 7739, 893, 899998} printf(1,"Number MDR MP\n") printf(1,"====== === ==\n") for i=1 to length(tests) do integer ti = tests[i] printf(1,"%6d %6d %6d\n",ti&mdr_mp(ti)) end for integer i=0, found = 0 sequence res = columnize({tagset(9,0)}) -- (ie {{0},{1},..,{9}}) while found<50 do -- (ie the full 10*5) integer mdr1 = mdr_mp(i)[1]+1 sequence m1 = res[mdr1] if length(m1)<6 then res[mdr1] = 0 -- (avoid p2js violation) m1 &= i res[mdr1] = m1 found += 1 end if i += 1 end while printf(1,"\nMDR 1 2 3 4 5") printf(1,"\n=== ===========================\n") for i=1 to 10 do printf(1,"%2d %5d %5d %5d %5d %5d\n",res[i]) end for
- Output:
Number MDR MP ====== === == 123321 8 3 7739 8 3 893 2 3 899998 0 2 MDR 1 2 3 4 5 === =========================== 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
Similar
with javascript_semantics function pdd(integer n) return sprintf("%2d",product(sq_sub(sprint(n),'0'))) end function printf(1,"Product of the decimal digits of 1..100:\n%s\n", {join_by(apply(tagset(100),pdd),1,10)})
- Output:
Product of the decimal digits of 1..100: 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 2 4 6 8 10 12 14 16 18 0 3 6 9 12 15 18 21 24 27 0 4 8 12 16 20 24 28 32 36 0 5 10 15 20 25 30 35 40 45 0 6 12 18 24 30 36 42 48 54 0 7 14 21 28 35 42 49 56 63 0 8 16 24 32 40 48 56 64 72 0 9 18 27 36 45 54 63 72 81 0
PL/I
version 1
multiple: procedure options (main); /* 29 April 2014 */
declare n fixed binary (31);
find_mdr: procedure;
declare (mdr, mp, p) fixed binary (31);
mdr = n;
do mp = 1 by 1 until (p <= 9);
p = 1;
do until (mdr = 0); /* Form product of the digits in mdr. */
p = mod(mdr, 10) * p;
mdr= mdr/10;
end;
mdr = p;
end;
put skip data (n, mdr, mp);
end find_mdr;
do n = 123321, 7739, 893, 899998;
call find_mdr;
end;
end multiple;- Output:
N= 123321 MDR= 8 MP= 3; N= 7739 MDR= 8 MP= 3; N= 893 MDR= 2 MP= 3; N= 899998 MDR= 0 MP= 2;
version 2
mdrt: Proc Options(main);
Dcl (x,p,r) Bin Fixed(31);
Put Edit('number persistence multiplicative digital root')(Skip,a);
Put Edit('------- ----------- ---------------------------')(Skip,a);
Call task1(123321);
Call task1( 7739);
Call task1( 893);
Call task1(899998);
task1: Procedure(x);
Dcl x Bin Fixed(31);
Call mdr(x,p,r);
Put Edit(x,p,r)(Skip,f(8),f(8),f(22));
End;
Dcl zn(0:9) Bin Fixed(31);
Dcl z(0:9,5) Bin Fixed(31);
zn=0;
zn(0)=1;
z(0,1)=0;
Do x=1 To 11111;
Call mdr(x,p,r);
If zn(r)<5 Then Do;
zn(r)+=1;
z(r,zn(r))=x;
End;
End;
Put Edit(' ')(Skip,a);
Put Edit('MDR first 5 numbers that have a matching MDR')(Skip,a);
Put Edit('--- ----------------------------------------')(Skip,a);
Do r=0 To 9;
Put Edit(r,' ')(Skip,f(3),a);
Do i=1 To 5;
Put Edit(z(r,i))(f(6));
End;
End;
mdr: Procedure(y,p,r);
Dcl (y,p,r) Bin Fixed(31);
Dcl (k,yy) Bin Fixed(31);
Dcl pic Pic'(10)9';
Dcl d Pic'9';
pic=abs(y);
Do p=1 By 1 Until(pic<10);
Do k=1 To 10 Until(substr(pic,k,1)>'0');
End;
r=1;
Do k=k To 10;
d=substr(pic,k,1);
r=r*d;
End;
pic=r;
End;
End;
End;- Output:
number persistence multiplicative digital root
------- ----------- ---------------------------
123321 3 8
7739 3 8
893 3 2
899998 2 0
MDR first 5 numbers that have a matching MDR
--- ----------------------------------------
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
Python
Python: Inspired by the solution to the Digital root task
try:
from functools import reduce
except:
pass
def mdroot(n):
'Multiplicative digital root'
mdr = [n]
while mdr[-1] > 9:
mdr.append(reduce(int.__mul__, (int(dig) for dig in str(mdr[-1])), 1))
return len(mdr) - 1, mdr[-1]
if __name__ == '__main__':
print('Number: (MP, MDR)\n====== =========')
for n in (123321, 7739, 893, 899998):
print('%6i: %r' % (n, mdroot(n)))
table, n = {i: [] for i in range(10)}, 0
while min(len(row) for row in table.values()) < 5:
mpersistence, mdr = mdroot(n)
table[mdr].append(n)
n += 1
print('\nMP: [n0..n4]\n== ========')
for mp, val in sorted(table.items()):
print('%2i: %r' % (mp, val[:5]))
- Output:
Number: (MP, MDR) ====== ========= 123321: (3, 8) 7739: (3, 8) 893: (3, 2) 899998: (2, 0) MP: [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]
Python: Inspired by the more efficient version of D.
Substitute the following function to run twice as fast when calculating mdroot(n) with n in range(1000000).
def mdroot(n):
count, mdr = 0, n
while mdr > 9:
m, digitsMul = mdr, 1
while m:
m, md = divmod(m, 10)
digitsMul *= md
mdr = digitsMul
count += 1
return count, mdr
- Output:
(Exactly the same as before).
Quackery
[ abs 1 swap
[ base share /mod
rot * swap
dup 0 = until ]
drop ] is digitproduct ( n --> n )
[ 0 swap
[ dup base share > while
dip 1+
digitproduct again ] ] is mdr ( n --> n n )
[ dup mdr
rot echo
say ": "
swap echo
say ", "
echo cr ] is task.1 ( n --> )
[ times
[ i^ [] swap dup rot
[ unrot dup mdr nip
swap dip
[ over = ]
swap iff
[ rot over join ]
else rot
dip 1+
dup size 5 = until ]
i^ echo say " : "
echo cr 2drop ] ] is task.2 ( n --> )
' [ 123321 7739 893 899998 ] witheach task.1
cr
10 task.2- Output:
123321: 3, 8 7739: 3, 8 893: 3, 2 899998: 2, 0 0 : [ 0 20 30 40 45 ] 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 ]
Racket
#lang racket
(define (digital-product n)
(define (inr-d-p m rv)
(cond
[(zero? m) rv]
[else (define-values (q r) (quotient/remainder m 10))
(if (zero? r) 0 (inr-d-p q (* rv r)))])) ; lazy on zero
(inr-d-p n 1))
(define (mdr/mp n)
(define (inr-mdr/mp m i)
(if (< m 10) (values m i) (inr-mdr/mp (digital-product m) (add1 i))))
(inr-mdr/mp n 0))
(printf "Number\tMDR\tmp~%======\t===\t==~%")
(for ((n (in-list '(123321 7739 893 899998))))
(define-values (mdr mp) (mdr/mp n))
(printf "~a\t~a\t~a~%" n mdr mp))
(printf "~%MDR\t[n0..n4]~%===\t========~%")
(for ((MDR (in-range 10)))
(define (has-mdr? n) (define-values (mdr mp) (mdr/mp n)) (= mdr MDR))
(printf "~a\t~a~%" MDR (for/list ((_ 5) (n (sequence-filter has-mdr? (in-naturals)))) n)))
- Output:
Number MDR mp ====== === == 123321 8 3 7739 8 3 893 2 3 899998 0 2 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)
Raku
(formerly Perl 6)
sub multiplicative-digital-root(Int $n) {
return .elems - 1, .[.end]
given cache($n, {[*] .comb} ... *.chars == 1)
}
for 123321, 7739, 893, 899998 {
say "$_: ", .&multiplicative-digital-root;
}
for ^10 -> $d {
say "$d : ", .[^5]
given (1..*).grep: *.&multiplicative-digital-root[1] == $d;
}
- Output:
123321: 3 8 7739: 3 8 893: 3 2 899998: 2 0 0 : 10 20 25 30 40 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
Red
Red ["Multiplicative digital root"]
mdr: function [
"Returns a block containing the mdr and persistence of an integer"
n [integer!]
][
persistence: 0
while [n > 10][
product: 1
m: n
while [m > 0][
product: m % 10 * product
m: to-integer m / 10
]
persistence: persistence + 1
n: product
]
reduce [n persistence]
]
foreach n [123321 7739 893 899998][
result: mdr n
print [pad n 6 "has multiplicative persistence" result/2 "and MDR" result/1]
]
print [newline "First five numbers with MDR of"]
repeat i 10 [
prin rejoin [i - 1 ": "]
hits: n: 0
while [hits < 5][
if i - 1 = first mdr n [
prin pad n 5
hits: hits + 1
]
n: n + 1
]
prin newline
]
- Output:
123321 has multiplicative persistence 3 and MDR 8 7739 has multiplicative persistence 3 and MDR 8 893 has multiplicative persistence 3 and MDR 2 899998 has multiplicative persistence 2 and MDR 0 First five numbers with MDR of 0: 0 20 30 40 45 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
REXX
idomatic version
/*REXX program finds the persistence and multiplicative digital root of some numbers.*/
numeric digits 100 /*increase the number of decimal digits*/
parse arg x /*obtain optional arguments from the CL*/
if x='' | x="," then x=123321 7739 893 899998 /*Not specified? Then use the default.*/
say center('number', 8) ' persistence multiplicative digital root'
say copies('─' , 8) ' ─────────── ───────────────────────────'
/* [↑] the title and separator. */
do j=1 for words(x); n=word(x, j) /*process each number in the X list.*/
parse value MDR(n) with mp mdr /*obtain the persistence and the MDR. */
say right(n,8) center(mp,13) center(mdr,30) /*display a number, persistence, MDR.*/
end /*j*/ /* [↑] show MP & MDR for each number. */
say copies('─' , 8) ' ─────────── ───────────────────────────'
say; say; target=5
say 'MDR first ' target " numbers that have a matching MDR"
say '═══ ═══════════════════════════════════════════════════'
do k=0 for 10; hits=0; _= /*show numbers that have an MDR of K. */
do m=k until hits==target /*find target numbers with an MDR of K.*/
if word( MDR(m), 2)\==k then iterate /*is this the MDR that's wanted? */
hits=hits + 1; _=space(_ m',') /*yes, we got a hit, add to the list. */
end /*m*/ /* [↑] built a list of MDRs that = K. */
say " "k': ['strip(_, , ',')"]" /*display the K (MDR) and the list. */
end /*k*/ /* [↑] done with the K MDR list. */
say '═══ ═══════════════════════════════════════════════════'
exit 0 /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
MDR: procedure; parse arg y; y=abs(y) /*get the number and determine the MDR.*/
do p=1 until y<10; parse var y r 2
do k=2 to length(y); r=r * substr(y, k, 1)
end /*k*/
y=r
end /*p*/ /* [↑] wash, rinse, and repeat ··· */
return p r /*return the persistence and the MDR. */
- output when using the default inputs:
number persistence multiplicative digital root
──────── ─────────── ───────────────────────────
123321 3 8
7739 3 8
893 3 2
899998 2 0
──────── ─────────── ───────────────────────────
MDR first 5 numbers that have a matching MDR
═══ ═══════════════════════════════════════════════════
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]
═══ ═══════════════════════════════════════════════════
ultra-fast version
This fast version can handle a target of five hundred numbers with ease for the 2nd part of the task's requirement.
/*REXX program finds the persistence and multiplicative digital root of some numbers.*/
numeric digits 2000 /*increase the number of decimal digits*/
parse arg target x /*obtain optional arguments from the CL*/
if \datatype(target, 'W') then target=25 /*Not specified? Then use the default.*/
if x='' | x="," then x=123321 7739 893 899998 /* " " " " " " */
say center('number',8) ' persistence multiplicative digital root'
say copies('─' ,8) ' ─────────── ───────────────────────────'
/* [↑] the title and the separator. */
do j=1 for words(x); n= abs( word(x, j) ) /*process each number in the list. */
parse value MDR(n) with mp mdr /*obtain the persistence and the MDR. */
say right(n,8) center(mp,13) center(mdr,30) /*display the number, persistence, MDR.*/
end /*j*/ /* [↑] show MP and MDR for each number.*/
say copies('─' ,8) ' ─────────── ───────────────────────────'
say; say /* [↓] show a blank and the title line.*/
say 'MDR first ' target " numbers that have a matching MDR"
say '═══ ' copies("═",(target+(target+1)**2)%2) /*display a separator line (for title).*/
do k=0 for 9; hits= 0 /*show numbers that have an MDR of K. */
_=
if k==7 then _= @7 /*handle the special case of seven. */
else do m=k until hits==target /*find target numbers with an MDR of K.*/
parse var m '' -1 ? /*obtain the right─most digit of M. */
if k\==0 then if ?==0 then iterate
if k==5 then if ?//2==0 then iterate
if k==1 then m= copies(1, hits+1)
else if MDR(m, 1)\==k then iterate
hits= hits + 1 /*got a hit, add to the list*/
_= space(_ m) /*elide superfluous blanks. */
if k==3 then do; o=strip(m, 'T', 1) /*strip trailing ones from M*/
if o==3 then m= copies(1, length(m))3 /*make a new M.*/
else do; t= pos(3, m) - 1 /*position of 3 */
m= overlay(3, translate(m, 1, 3), t)
end /* [↑] shift the "3" 1 place left.*/
m= m - 1 /*adjust for DO index increment.*/
end /* [↑] a shortcut to adj DO index*/
end /*m*/ /* [↑] built a list of MDRs = K */
say " "k': ['_"]" /*display the K (MDR) and the list. */
if k==3 then @7= translate(_, 7, k) /*save for later, a special "7" case.*/
end /*k*/ /* [↑] done with the K MDR list. */
@.= /* [↓] handle MDR of "9" special. */
_= translate(@7, 9, 7) /*translate string for MDR of nine. */
@9= translate(_, , ',') /*remove trailing commas from numbers. */
@3= /*assign null string before building. */
do j=1 for words(@9) /*process each number for MDR 9 case.*/
_= space( translate( word(@9, j), , 9), 0) /*elide all "9"s using SPACE(x,0).*/
L= length(_) + 1 /*use a "fudged" length of the number. */
new= /*these are the new numbers (so far). */
do k=0 for L; q= insert(3, _, k) /*insert the 1st "3" into the number*/
do i=k to L; z= insert(3, q, i) /* " " 2nd "3" " " " */
if @.z\=='' then iterate /*if already define, ignore the number.*/
@.z= z; new= z new /*define it, and then add to the list.*/
end /*i*/ /* [↑] end of 2nd insertion of "3".*/
end /*k*/ /* [↑] " " 1st " " " */
@3= space(@3 new) /*remove blanks, then add to the list.*/
end /*j*/ /* [↑] end of insertion of the "3"s. */
@= /* [↓] merge two lists, 3s and 9s. */
a1= @9; a2= @3 /*define some strings for the merge. */
do while a1\=='' & a2\=='' /*process while the lists aren't empty.*/
x= word(a1, 1); y= word(a2, 1) /*obtain the 1st word in A1 & A2 lists.*/
if x=='' | y=='' then leave /*are X or Y empty? */
if x<y then do; @= @ x; a1= delword(a1, 1, 1); end /*add X to the @ list*/
else do; @= @ y; a2= delword(a2, 1, 1); end /* " Y " " " " */
end /*while*/ /* [↑] only process just enough nums. */
@= subword(@, 1, target) /*elide the last trailing comma in list*/
say " "9': ['@"]" /*display the "9" (MDR) and the list.*/
say '═══ ' copies("═",(target+(target+1)**2)%2) /*display a separator line (for title).*/
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
MDR: procedure; parse arg y,s; y= abs(y) /*get the number and determine the MDR.*/
do p=1 until y<10; parse var y r 2
do k=2 to length(y); r= r * substr(y, k, 1)
end /*k*/
y= r
end /*p*/ /* [↑] wash, rinse, and repeat ··· */
if s==1 then return r /*return multiplicative digital root. */
return p r /*return the persistence and the MDR. */
- output when using the input of: 34
number persistence multiplicative digital root
──────── ─────────── ───────────────────────────
123321 3 8
7739 3 8
893 3 2
899998 2 0
──────── ─────────── ───────────────────────────
MDR first 34 numbers that have a matching MDR
═══ ═════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════
0: [0 10 20 25 30 40 45 50 52 54 55 56 58 59 60 65 69 70 78 80 85 87 90 95 96 100 101 102 103 104 105 106 107 108]
1: [1 11 111 1111 11111 111111 1111111 11111111 111111111 1111111111 11111111111 111111111111 1111111111111 11111111111111 111111111111111 1111111111111111 11111111111111111 111111111111111111 1111111111111111111 11111111111111111111 111111111111111111111 1111111111111111111111 11111111111111111111111 111111111111111111111111 1111111111111111111111111 11111111111111111111111111 111111111111111111111111111 1111111111111111111111111111 11111111111111111111111111111 111111111111111111111111111111 1111111111111111111111111111111 11111111111111111111111111111111 111111111111111111111111111111111 1111111111111111111111111111111111]
2: [2 12 21 26 34 37 43 62 73 112 121 126 134 137 143 162 173 211 216 223 232 261 278 279 287 297 299 314 317 322 341 367 369 371]
3: [3 13 31 113 131 311 1113 1131 1311 3111 11113 11131 11311 13111 31111 111113 111131 111311 113111 131111 311111 1111113 1111131 1111311 1113111 1131111 1311111 3111111 11111113 11111131 11111311 11113111 11131111 11311111]
4: [4 14 22 27 39 41 72 89 93 98 114 122 127 139 141 172 189 193 198 212 217 221 249 266 271 277 294 319 333 338 346 364 379 383]
5: [5 15 35 51 53 57 75 115 135 151 153 157 175 315 351 355 359 395 511 513 517 531 535 539 553 557 571 575 579 593 597 715 751 755]
6: [6 16 23 28 32 44 47 48 61 68 74 82 84 86 116 123 128 132 144 147 148 161 168 174 182 184 186 213 218 224 227 228 231 238]
7: [7 17 71 117 171 711 1117 1171 1711 7111 11117 11171 11711 17111 71111 111117 111171 111711 117111 171111 711111 1111117 1111171 1111711 1117111 1171111 1711111 7111111 11111117 11111171 11111711 11117111 11171111 11711111]
8: [8 18 24 29 36 38 42 46 49 63 64 66 67 76 77 79 81 83 88 92 94 97 99 118 124 129 136 138 142 146 149 163 164 166]
9: [9 19 33 91 119 133 191 313 331 911 1119 1133 1191 1313 1331 1911 3113 3131 3311 9111 11119 11133 11191 11313 11331 11911 13113 13131 13311 19111 31113 31131 31311 33111]
═══ ═════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════════
Similar
/*REXX pgm finds positive integers when shown in hex that can't be written with dec digs*/
parse arg n cols . /*obtain optional argument from the CL.*/
if n=='' | n=="," then n = 100 /*Not specified? Then use the default.*/
if cols=='' | cols=="," then cols= 10 /* " " " " " " */
w= 10 /*width of a number in any column. */
title= ' the product of the decimal digits of N, where N < ' n
say ' index │'center(title, 1 + cols*(w+1) ) /*display the title for the output. */
say '───────┼'center("" , 1 + cols*(w+1), '─') /* " a sep " " " */
$=; idx= 1 /*list of products (so far); IDX=index.*/
do #=1 for n; L= length(#) /*find products of the dec. digs of J. */
p= left(#, 1) /*use first digit as the product so far*/
do j=2 for L-1 until p==0 /*add an optimization when product is 0*/
p= p * substr(#, j, 1) /*multiply the product by the next dig.*/
end /*j*/
$= $ right(p, w) /*add the product ───► the $ list. */
if #//cols \== 0 then iterate /*have we populated a line of output? */
say center(idx, 7)'│' substr($, 2); $= /*display what we have so far (cols). */
idx= idx + cols /*bump the index count for the output*/
end /*#*/ /*stick a fork in it, we're all done. */
if $\=='' then say center(idx, 7)"│" substr($, 2) /*possible display residual output.*/
say '───────┴'center("" , 1 + cols*(w+1), '─') /*display the foot sep for output. */
- output when using the default inputs:
index │ the product of the decimal digits of N, where N < 100 ───────┼─────────────────────────────────────────────────────────────────────────────────────────────────────────────── 1 │ 1 2 3 4 5 6 7 8 9 0 11 │ 1 2 3 4 5 6 7 8 9 0 21 │ 2 4 6 8 10 12 14 16 18 0 31 │ 3 6 9 12 15 18 21 24 27 0 41 │ 4 8 12 16 20 24 28 32 36 0 51 │ 5 10 15 20 25 30 35 40 45 0 61 │ 6 12 18 24 30 36 42 48 54 0 71 │ 7 14 21 28 35 42 49 56 63 0 81 │ 8 16 24 32 40 48 56 64 72 0 91 │ 9 18 27 36 45 54 63 72 81 0 ───────┴───────────────────────────────────────────────────────────────────────────────────────────────────────────────
Ring
# Project : Digital root/Multiplicative digital root
load "stdlib.ring"
root = newlist(10, 5)
for r = 1 to 10
for x = 1 to 5
root[r][x] = 0
next
next
root2 = list(10)
for y = 1 to 10
root2[y] = 0
next
see "Number MDR MP" + nl
num = [123321, 7739, 893, 899998]
digroot(num)
see nl
num = 0:12000
digroot(num)
see "First five numbers with MDR in first column:" + nl
for n1 = 1 to 10
see "" + (n1-1) + " => "
for n2 = 1 to 5
see "" + root[n1][n2] + " "
next
see nl
next
func digroot(num)
for n = 1 to len(num)
sum = 0
numold = num[n]
while true
pro = 1
strnum = string(numold)
for nr = 1 to len(strnum)
pro = pro * number(strnum[nr])
next
sum = sum + 1
numold = pro
numn = string(num[n])
sp = 6 - len(string(num[n]))
if sp > 0
for p = 1 to sp + 2
numn = " " + numn
next
ok
if len(string(numold)) = 1 and len(num) < 5
see "" + numn + " " + numold + " " + sum + nl
exit
ok
if len(string(numold)) = 1 and len(num) > 4
root2[numold+1] = root2[numold+1] + 1
if root2[numold+1] < 6
root[numold+1][root2[numold+1]] = num[n]
ok
exit
ok
end
nextOutput:
Number MDR MP 123321 8 3 7739 8 3 893 2 3 899998 0 2 First five numbers with MDR in first column: 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
Similar
load "stdlib.ring"
see "working..." + nl
see "Product of decimal digits of n:" + nl
row = 0
limit = 100
for n = 1 to limit
prod = 1
strn = string(n)
for m = 1 to len(strn)
prod = prod * number(strn[m])
next
see "" + prod + " "
row = row + 1
if row%5 = 0
see nl
ok
next
see "done..." + nl- Output:
working... Product of decimal digits of n: 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 2 4 6 8 10 12 14 16 18 0 3 6 9 12 15 18 21 24 27 0 4 8 12 16 20 24 28 32 36 0 5 10 15 20 25 30 35 40 45 0 6 12 18 24 30 36 42 48 54 0 7 14 21 28 35 42 49 56 63 0 8 16 24 32 40 48 56 64 72 0 9 18 27 36 45 54 63 72 81 0 done...
RPL
≪ 1 SWAP
DO 10 / LAST MOD ROT * RND SWAP FLOOR
UNTIL DUP NOT END DROP
≫ 'MDGIT' STO
≪ 0 WHILE OVER 9 > REPEAT
1 + SWAP MDGIT SWAP END SWAP R→C
≫ 'MDPR' STO
≪ { 123321 7739 893 899998 } → cases
≪ {} 1 cases SIZE FOR j cases j GET MDPR + NEXT
≫ ≫ 'TASK1' STO
≪ 1 10 START { 0 0 0 0 0 } NEXT 10 →LIST 'tab' STO 50 'cnt' STO
1 99999 FOR j
j MDPR IM 1 + tab OVER GET
IF DUP 0 POS THEN
LAST j PUT 'tab' ROT ROT PUT cnt 1 -
IF DUP THEN 'cnt' STO ELSE 99999 'j' STO END
ELSE DROP2 END
NEXT tab
≫ 'TASK2' STO
- Output:
2: { (3,8) (3,8) (3,2) (2,0) }
1: { { 10 20 25 30 40 }
{ 1 11 111 1111 11111 }
{ 2 12 21 26 34 }
{ 3 13 31 113 131 }
{ 4 14 22 27 39 }
{ 5 15 35 51 53 }
{ 6 16 23 28 32 }
{ 7 17 71 117 171 }
{ 8 18 24 29 36 }
{ 9 19 33 91 119 } }
Ruby
def mdroot(n)
mdr, persist = n, 0
until mdr < 10 do
mdr = mdr.digits.inject(:*)
persist += 1
end
[mdr, persist]
end
puts "Number: MDR MP", "====== === =="
[123321, 7739, 893, 899998].each{|n| puts "%6d: %d %2d" % [n, *mdroot(n)]}
counter = Hash.new{|h,k| h[k]=[]}
0.step do |i|
counter[mdroot(i).first] << i
break if counter.values.all?{|v| v.size >= 5 }
end
puts "", "MDR: [n0..n4]", "=== ========"
10.times{|i| puts "%3d: %p" % [i, counter[i].first(5)]}
- Output:
Number: MDR MP ====== === == 123321: 8 3 7739: 8 3 893: 2 3 899998: 0 2 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]
Rust
// Multiplicative digital root
fn mdroot(n: u32) -> (u32, u32) {
let mut count = 0;
let mut mdr = n;
while mdr > 9 {
let mut m = mdr;
let mut digits_mul = 1;
while m > 0 {
digits_mul *= m % 10;
m /= 10;
}
mdr = digits_mul;
count += 1;
}
return (count, mdr);
}
fn main() {
println!("Number: (MP, MDR)\n====== =========");
for n in [123321, 7739, 893, 899998] {
println!("{:6}: {:?}", n, mdroot(n));
}
let mut table = vec![vec![0_u32; 0]; 10];
let mut n = 0;
while table.iter().map(|row| row.len()).min().unwrap() < 5 {
let (_, mdr) = mdroot(n);
table[mdr as usize].push(n);
n += 1;
}
println!("\nMDR First 5 with matching MDR\n=== =========================");
table.sort();
for a in table {
println!("{:2}: {:5}{:6}{:6}{:6}{:6}", a[0], a[0], a[1], a[2], a[3], a[4]);
}
}
- Output:
Number: (MP, MDR) ====== ========= 123321: (3, 8) 7739: (3, 8) 893: (3, 2) 899998: (2, 0) MDR First 5 with matching MDR === ========================= 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
Scala
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(", "))}
}
- Output:
Number MDR MP
====== === ==
123321 8 3
7739 8 3
893 2 3
899998 0 2
393900588225 0 1
999999999999 0 3
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]
Scheme
; Convert an integer into a list of its digits.
(define integer->list
(lambda (integer)
(let loop ((list '()) (int integer))
(if (< int 10)
(cons int list)
(loop (cons (remainder int 10) list) (quotient int 10))))))
; Return the product of the digits of an integer.
(define integer-product-digits
(lambda (integer)
(fold-left * 1 (integer->list integer))))
; Compute the multiplicative digital root and multiplicative persistence of an integer.
; Return as a cons of (mdr . mp).
(define mdr-mp
(lambda (integer)
(let loop ((int integer) (cnt 0))
(if (< int 10)
(cons int cnt)
(loop (integer-product-digits int) (1+ cnt))))))
; Emit a table of integer, multiplicative digital root, and multiplicative persistence
; for the example integers given. Example list ends with sequence A003001 from OEIS.
(printf "~16@a ~6@a ~6@a~%" "Integer" "Root" "Pers.")
(printf "~16@a ~6@a ~6@a~%" "===============" "======" "======")
(let rowloop ((intlist '(123321 7739 893 899998
0 10 25 39 77 679 6788 68889 2677889 26888999 3778888999 277777788888899)))
(when (pair? intlist)
(let* ((int (car intlist))
(mm (mdr-mp int)))
(printf "~16@a ~6@a ~6@a~%" int (car mm) (cdr mm))
(rowloop (cdr intlist)))))
; Emit a table of multiplicative digital root versus the first five integers having that MDR.
(newline)
(printf "~5@a ~a~%" "Root" "First five integers with that root")
(printf "~5@a ~a~%" "====" "==================================")
(let ((mdrslsts (make-vector 10 '())))
(do ((integer 0 (1+ integer)))
((>= (fold-left min 5 (vector->list (vector-map length mdrslsts))) 5))
(let ((mdr (car (mdr-mp integer))))
(when (< (length (vector-ref mdrslsts mdr)) 5)
(vector-set! mdrslsts mdr (append (vector-ref mdrslsts mdr) (list integer))))))
(do ((mdr 0 (1+ mdr)))
((>= mdr 10))
(printf "~5@a" mdr)
(for-each (lambda (int) (printf "~7@a" int)) (vector-ref mdrslsts mdr))
(newline)))
- Output:
Integer Root Pers.
=============== ====== ======
123321 8 3
7739 8 3
893 2 3
899998 0 2
0 0 0
10 0 1
25 0 2
39 4 3
77 8 4
679 6 5
6788 0 6
68889 0 7
2677889 0 8
26888999 0 9
3778888999 0 10
277777788888899 0 11
Root First five integers with that root
==== ==================================
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
Sidef
func mdroot(n) {
var (mdr, persist) = (n, 0)
while (mdr >= 10) {
mdr = mdr.digits.prod
++persist
}
[mdr, persist]
}
say "Number: MDR MP\n====== === =="
[123321, 7739, 893, 899998].each{|n| "%6d: %3d %3d\n" \
.printf(n, mdroot(n)...) }
var counter = Hash()
Inf.times { |j|
counter{mdroot(j).first} := [] << j
break if counter.values.all {|v| v.len >= 5 }
}
say "\nMDR: [n0..n4]\n=== ========"
10.times {|i| "%3d: %s\n".printf(i, counter{i}.first(5)) }
- Output:
Number: MDR MP ====== === == 123321: 8 3 7739: 8 3 893: 2 3 899998: 0 2 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]
Tcl
proc mdr {n} {
if {$n < 0 || ![string is integer $n]} {
error "must be an integer"
}
for {set i 0} {$n > 9} {incr i} {
set n [tcl::mathop::* {*}[split $n ""]]
}
return [list $i $n]
}
Demonstrating:
puts "Number: MP MDR"
puts [regsub -all . "Number: MP MDR" -]
foreach n {123321 7739 893 899998} {
puts [format "%6d: %2d %3d" $n {*}[mdr $n]]
}
puts ""
# The longEnough variable counts how many roots have at least 5 values accumulated for them
for {set i [set longEnough 0]} {$longEnough < 10} {incr i} {
set root [lindex [mdr $i] 1]
if {[llength [lappend accum($root) $i]] == 5} {incr longEnough}
}
puts "MDR: \[n\u2080\u2026n\u2084\]"
puts [regsub -all . "MDR: \[n\u2080\u2026n\u2084\]" -]
for {set i 0} {$i < 10} {incr i} {
puts [format "%3d: (%s)" $i [join [lrange $accum($i) 0 4] ", "]]
}
- Output:
Number: MP MDR -------------- 123321: 3 8 7739: 3 8 893: 3 2 899998: 2 0 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)
V (Vlang)
// Only valid for n > 0 && base >= 2
fn mult(nn u64, base int) u64 {
mut n := nn
mut mult := u64(0)
for mult = 1; mult > 0 && n > 0; n /= u64(base) {
mult *= n % u64(base)
}
return mult
}
// Only valid for n >= 0 && base >= 2
fn multi_digital_root(n u64, base int) (int, int) {
mut m := u64(0)
mut mp := 0
for m = n; m >= u64(base); mp++ {
m = mult(m, base)
}
return mp, int(m)
}
const base = 10
fn main() {
size := 5
println("${'Number':20} ${'MDR':3} ${'MP':3}")
for n in [
u64(123321), 7739, 893, 899998,
18446743999999999999,
// From http://mathworld.wolfram.com/MultiplicativePersistence.html
3778888999, 277777788888899,
] {
mp, mdr := multi_digital_root(n, base)
println("${n:20} ${mdr:3} ${mp:3}")
}
println('')
mut list := [base][]u64{init: []u64{len: 0, cap:size}}
for cnt, n := size*base, u64(0); cnt > 0; n++ {
_, mdr := multi_digital_root(n, base)
if list[mdr].len < size {
list[mdr] << n
cnt--
}
}
println("${'MDR':3}: First")
for i, l in list {
println("${i:3}: $l")
}
}- Output:
Number MDR MP
123321 8 3
7739 8 3
893 2 3
899998 0 2
18446743999999999999 0 2
3778888999 0 10
277777788888899 0 11
MDR: First
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]
Wren
The size of some of the numbers here is such that we need to use BigInt.
import "./big" for BigInt
import "./fmt" for Fmt
// Only valid for n > 0 && base >= 2
var mult = Fn.new { |n, base|
var m = BigInt.one
while (m > BigInt.zero && n > BigInt.zero) {
var dm = n.divMod(base)
m = m * dm[1]
n = dm[0]
}
return m
}
// Only valid for n >= 0 && base >= 2
var multDigitalRoot = Fn.new { |n, base|
base = BigInt.new(base)
var m = n.copy()
var mp = BigInt.zero
while (m >= base) {
m = mult.call(m, base)
mp = mp.inc
}
return [mp, m.toSmall]
}
var base = 10
var size = 5
var tests = [
123321, 7739, 893, 899998,"18446743999999999999", 3778888999, "277777788888899"
]
var testFmt = "$20s $3s $3s"
Fmt.print(testFmt, "Number", "MDR", "MP")
for (test in tests) {
var n = BigInt.new(test)
var mpdr = multDigitalRoot.call(n, base)
Fmt.print(testFmt, n, mpdr[1], mpdr[0])
}
System.print()
var list = List.filled(base, null)
for (i in 0...base) list[i] = []
var cnt = size * base
var n = BigInt.zero
while (cnt > 0) {
var mpdr = multDigitalRoot.call(n, base)
var mdr = mpdr[1]
if (list[mdr].count < size) {
list[mdr].add(n)
cnt = cnt - 1
}
n = n.inc
}
Fmt.print("$3s: $s", "MDR", "First")
var i = 0
for (l in list) {
Fmt.print("$3d: $s", i, l.toString)
i = i + 1
}
- Output:
Number MDR MP
123321 8 3
7739 8 3
893 2 3
899998 0 2
18446743999999999999 0 2
3778888999 0 10
277777788888899 0 11
MDR: First
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]
XPL0
\Calculate the Multiplicative Digital Root (MDR) and
\ Multiplicative Persistence (MP) of N
procedure GetMDR ( N, MDR, MP );
integer N, MDR, MP, V;
begin
MP(0) := 0;
MDR(0) := abs( N );
while MDR(0) > 9 do begin
V := MDR(0);
MDR(0) := 1;
repeat
MDR(0) := MDR(0) * rem( V / 10 );
V := V / 10;
until V = 0;
MP(0) := MP(0) + 1;
end; \while_mdr_gt_9
end; \GetMDR
define RequiredMDRs = 5;
integer FirstFew ( 9+1, 1+RequiredMDRs );
integer MDRFound ( 9+1 );
integer TotalFound, FoundPos, RequiredTotal, N, I, V, L;
integer MDR, MP;
begin
\task test cases
Text(0, " N MDR MP^m^j" );
L := [ 123321, 7739, 893, 899998 ];
for N := 0 to 3 do begin
GetMDR( L(N), @MDR, @MP );
Format(8, 0); RlOut(0, float(L(N)));
Format(4, 0); RlOut(0, float(MDR));
Format(3, 0); RlOut(0, float(MP));
CrLf(0)
end; \for_N
\find the first 5 numbers with each possible MDR
begin
for I := 0 to 9 do MDRFound( I ) := 0;
TotalFound := 0;
RequiredTotal := 10 * RequiredMDRs;
N := -1;
while TotalFound < RequiredTotal do begin
N := N + 1;
GetMDR( N, @MDR, @MP );
if MDRFound( MDR ) < RequiredMDRs then begin
\Found another number with this MDR and haven't found enough
TotalFound := TotalFound + 1;
MDRFound( MDR ) := MDRFound( MDR ) + 1;
FirstFew( MDR, MDRFound( MDR ) ) := N
end \if_Found_another_MDR
end; \while_TotalFound_lt_RequiredTotal
\print the table of MDRs and numbers
Text(0, "MDR: [N0..N4]^m^j" );
Text(0, "=== ========^m^j" );
for V := 0 to 9 do begin
ChOut(0, ^ ); IntOut(0, V); Text(0, ": [");
for FoundPos := 1 to RequiredMDRs do begin
if FoundPos > 1 then Text( 0, ", " );
IntOut( 0, FirstFew( V, FoundPos ) )
end; \for_FoundPos
Text(0, "]^m^j")
end \for_v
end
end- Output:
N MDR MP
123321 8 3
7739 8 3
893 2 3
899998 0 2
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]
zkl
fcn mdroot(n){ // Multiplicative digital root
mdr := List(n);
while (mdr[-1] > 9){
mdr.append(mdr[-1].split().reduce('*,1));
}
return(mdr.len() - 1, mdr[-1]);
}fcn mdroot(n){
count:=0; mdr:=n;
while(mdr > 9){
m:=mdr; digitsMul:=1;
while(m){
reg md;
m,md=m.divr(10);
digitsMul *= md;
}
mdr = digitsMul;
count += 1;
}
return(count, mdr);
}println("Number: (MP, MDR)\n======= =========");
foreach n in (T(123321, 7739, 893, 899998))
{ println("%7,d: %s".fmt(n, mdroot(n))) }
table:=D([0..9].zip(fcn{List()}).walk()); // dictionary(0:List, 1:List, ...)
n :=0;
while(table.values.filter(fcn(r){r.len()<5})){ // until each entry has >=5 values
mpersistence, mdr := mdroot(n);
table[mdr].append(n);
n += 1;
}
println("\nMP: [n0..n4]\n== ========");
foreach mp in (table.keys.sort()){
println("%2d: %s".fmt(mp, table[mp][0,5])); //print first five values
}- Output:
Number: (MP, MDR)
======= =========
123,321: L(3,8)
7,739: L(3,8)
893: L(3,2)
899,998: L(2,0)
MP: [n0..n4]
== ========
0: L(0,10,20,25,30)
1: L(1,11,111,1111,11111)
2: L(2,12,21,26,34)
3: L(3,13,31,113,131)
4: L(4,14,22,27,39)
5: L(5,15,35,51,53)
6: L(6,16,23,28,32)
7: L(7,17,71,117,171)
8: L(8,18,24,29,36)
9: L(9,19,33,91,119)
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