Digital root/Multiplicative digital root
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.
- 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
<lang 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])</lang>
- 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]].
<lang Ada>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;</lang>
- 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
<lang algol68># Multiplicative Digital Roots #
- structure to hold the results of calculating the digital root & persistence #
MODE DR = STRUCT( INT root, INT persistence );
- calculate the multiplicative digital root and persistence of a number #
PROC md root = ( INT number )DR: BEGIN
# calculate the product of the digits of a number # PROC digit product = ( 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; # digit product #
INT mp := 0; INT mdr := ABS number;
WHILE mdr > 9
DO
mp +:= 1;
mdr := digit product( mdr )
OD;
( mdr, mp )
END; # md root #
- prints a number and its MDR and MP #
PROC print md root = ( INT number )VOID: BEGIN
DR mdr = md root( 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 = md root( 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 #
main:(
print md root( 123321 ); print md root( 7739 ); print md root( 893 ); print md root( 899998 ); tabulate mdr( 5 )
)</lang>
- 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
<lang algolw>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.</lang>
- 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
<lang 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</lang>
- 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
<lang 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 )
);</lang>
- 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
<lang 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;
} </lang>
- 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#
<lang csharp>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]));
}
}</lang>
- 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++
<lang cpp>
- 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" );
} </lang>
- 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 | +-------+------------------------------------+
Common Lisp
<lang 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))</lang>
- 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
<lang oberon2> 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. </lang> 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
<lang 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));
}</lang>
- 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
<lang d>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));
}</lang>
More Efficient Version
<lang d>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));
}</lang> The output is similar.
Elixir
<lang 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</lang>
- 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]
Factor
<lang 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</lang>
- 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
<lang 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
</lang>
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
<lang 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</lang>
- 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
<lang 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) } }</lang>
- 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.
<lang haskell>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</lang>
- 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: <lang unicon>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</lang>
- 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:
<lang J> 10&#.inv 123321 1 2 3 3 2 1</lang>
Second, we need to find their product:
<lang J> */@(10&#.inv) 123321 36</lang>
Then we use this inductively until it converges:
<lang J> */@(10&#.inv)^:a: 123321 123321 36 18 8</lang>
MP is one less than the length of this list, and MDR is the last element of this list:
<lang J> (<:@#,{:) */@(10&#.inv)^:a: 123321 3 8
(<:@#,{:) */@(10&#.inv)^:a: 7739
3 8
(<:@#,{:) */@(10&#.inv)^:a: 893
3 2
(<:@#,{:) */@(10&#.inv)^:a: 899998
2 0</lang>
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):
<lang J> (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</lang>
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
<lang 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};
}
}</lang>
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
<lang 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;</lang>
Example:<lang jq> 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($i))"),
"", "Tabulation", "MDR: [n0..n4]", (tabulate(5) | neatly)</lang>
- Output:
<lang sh>$ 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]</lang>
Julia
Function <lang Julia> 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 </lang> Main <lang Julia> 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 </lang>
- 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
<lang scala>// 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()
}
}</lang>
- 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
<lang mathematica> 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[numsi + 1, 5]}, {i, 0, 9}],
TableHeadings -> {None, {"MDR", "First 5"}}, TableDepth -> 2]
</lang>
- 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
<lang nim>import strutils, future
template newSeqWith(len: int, init: expr): expr =
var result {.gensym.} = newSeq[type(init)](len)
for i in 0 .. <len:
result[i] = init
result
proc mdroot(n): 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]</lang>
- 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
<lang parigp>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))))</lang>
- 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]]
Perl
<lang 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";
}</lang>
- 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]
Perl 6
<lang perl6>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;
}</lang>
- 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
Phix
<lang Phix>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 = repeat({},10) while found<50 do
integer {mdr,mp} = mdr_mp(i)
if length(res[mdr+1])<5 then
res[mdr+1] &= i
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",prepend(res[i],i-1))
end for</lang>
- 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
PL/I
version 1
<lang pli>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;</lang>
- 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
<lang pli> 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;</lang>
- 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
<lang python>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]))</lang>
- 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). <lang python>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</lang>
- Output:
(Exactly the same as before).
Racket
<lang 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)))</lang>
- 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)
REXX
idomatic version
<lang rexx>/*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; 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. */
exit /*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. */</lang>
- 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. <lang rexx>/*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 /* [↓] 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. */
a1=@9; a2=@3 /*define some strings for the merge. */ @= /* [↓] merge two lists, 3s and 9s. */
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.*/ 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. */</lang>
- 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]
Ring
<lang 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
next
</lang> Output:
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
Ruby
<lang ruby>def mdroot(n)
mdr, persist = n, 0 until mdr < 10 do mdr = mdr.to_s.each_char.map(&:to_i).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)]}</lang>
- 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]
Scala
<lang 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(", "))}
}</lang>
- 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]
Sidef
<lang ruby>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)) }</lang>
- 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
<lang 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]
}</lang> Demonstrating: <lang tcl>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] ", "]]
}</lang>
- 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)
zkl
<lang 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]);
}</lang> <lang zkl>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);
}</lang> <lang zkl>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
}</lang>
- 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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