Smith numbers
Smith numbers are numbers such that the sum of the decimal digits of the integers that make up that number is the same as the sum of the decimal digits of its prime factors excluding 1.
By definition, all primes are excluded as they (naturally) satisfy this condition!
Smith numbers are also known as joke numbers.
- Example
Using the number 166
Find the prime factors of 166 which are: 2 x 83
Then, take those two prime factors and sum all their decimal digits: 2 + 8 + 3 which is 13
Then, take the decimal digits of 166 and add their decimal digits: 1 + 6 + 6 which is 13
Therefore, the number 166 is a Smith number.
- Task
Write a program to find all Smith numbers below 10000.
- See also
- from Wikipedia: [Smith number].
- from MathWorld: [Smith number].
- from OEIS A6753: [OEIS sequence A6753].
- from OEIS A104170: [Number of Smith numbers below 10^n].
- from The Prime pages: [Smith numbers].
ALGOL 68
<lang algol68># sieve of Eratosthene: sets s[i] to TRUE if i is prime, FALSE otherwise # PROC sieve = ( REF[]BOOL s )VOID:
BEGIN
# start with everything flagged as prime #
FOR i TO UPB s DO s[ i ] := TRUE OD;
# sieve out the non-primes #
s[ 1 ] := FALSE;
FOR i FROM 2 TO ENTIER sqrt( UPB s ) DO
IF s[ i ] THEN FOR p FROM i * i BY i TO UPB s DO s[ p ] := FALSE OD FI
OD
END # sieve # ;
- construct a sieve of primes up to the maximum number required for the task #
INT max number = 10 000; [ 1 : max number ]BOOL is prime; sieve( is prime );
- returns the sum of the digits of n #
OP DIGITSUM = ( INT n )INT:
BEGIN
INT sum := 0;
INT rest := ABS n;
WHILE rest > 0 DO
sum +:= rest MOD 10;
rest OVERAB 10
OD;
sum
END # DIGITSUM # ;
- returns TRUE if n is a Smith number, FALSE otherwise #
- n must be between 1 and max number #
PROC is smith = ( INT n )BOOL:
IF is prime[ ABS n ] THEN
# primes are not Smith numbers #
FALSE
ELSE
# find the factors of n and sum the digits of the factors #
INT rest := ABS n;
INT factor digit sum := 0;
INT factor := 2;
WHILE factor < max number AND rest > 1 DO
IF NOT is prime[ factor ] THEN
# factor isn't a prime #
factor +:= 1
ELSE
# factor is a prime #
IF rest MOD factor /= 0 THEN
# factor isn't a factor of n #
factor +:= 1
ELSE
# factor is a factor of n #
rest OVERAB factor;
factor digit sum +:= DIGITSUM factor
FI
FI
OD;
( factor digit sum = DIGITSUM n )
FI # is smith # ;
- print all the Smith numbers below the maximum required #
INT smith count := 0; FOR n TO max number - 1 DO
IF is smith( n ) THEN
# have a smith number #
print( ( whole( n, -7 ) ) );
smith count +:= 1;
IF smith count MOD 10 = 0 THEN
print( ( newline ) )
FI
FI
OD; print( ( newline, "THere are ", whole( smith count, -7 ), " Smith numbers below ", whole( max number, -7 ), newline ) ) </lang>
- Output:
4 22 27 58 85 94 121 166 202 265
274 319 346 355 378 382 391 438 454 483
...
9717 9735 9742 9760 9778 9840 9843 9849 9861 9880
9895 9924 9942 9968 9975 9985
THere are 376 Smith numbers below 10000
C++
<lang cpp>
- include <iostream>
- include <vector>
- include <iomanip>
void primeFactors( unsigned n, std::vector<unsigned>& r ) {
int f = 2; if( n == 1 ) r.push_back( 1 );
else {
while( true ) {
if( !( n % f ) ) {
r.push_back( f );
n /= f; if( n == 1 ) return;
}
else f++;
}
}
} unsigned sumDigits( unsigned n ) {
unsigned sum = 0, m;
while( n ) {
m = n % 10; sum += m;
n -= m; n /= 10;
}
return sum;
} unsigned sumDigits( std::vector<unsigned>& v ) {
unsigned sum = 0;
for( std::vector<unsigned>::iterator i = v.begin(); i != v.end(); i++ ) {
sum += sumDigits( *i );
}
return sum;
} void listAllSmithNumbers( unsigned n ) {
std::vector<unsigned> pf;
for( unsigned i = 4; i < n; i++ ) {
primeFactors( i, pf ); if( pf.size() < 2 ) continue;
if( sumDigits( i ) == sumDigits( pf ) )
std::cout << std::setw( 4 ) << i << " ";
pf.clear();
}
std::cout << "\n\n";
} int main( int argc, char* argv[] ) {
listAllSmithNumbers( 10000 ); return 0;
} </lang>
- Output:
4 22 27 58 85 94 121 166 202 265 274 319 346 355 378 382 391 438 454 483 517 526 535 562 576 627 634 636 645 663 666 690 ... 9301 9330 9346 9355 9382 9386 9387 9396 9427 9483 9535 9571 9598 9633 9634 9639 9648 9657 9684 9708 9717 9735 9742 9760 9778 9843 9849 9861 9880 9895 9975 9985
Elixir
<lang elixir>defmodule Smith do
def number?(n) do d = decomposition(n) length(d)>1 and sum_digits(n) == Enum.map(d, &sum_digits/1) |> Enum.sum end defp sum_digits(n) do Integer.digits(n) |> Enum.sum end defp decomposition(n, k\\2, acc\\[]) defp decomposition(n, k, acc) when n < k*k, do: [n | acc] defp decomposition(n, k, acc) when rem(n, k) == 0, do: decomposition(div(n, k), k, [k | acc]) defp decomposition(n, k, acc), do: decomposition(n, k+1, acc)
end
m = 10000 smith = Enum.filter(1..m, &Smith.number?/1) IO.puts "#{length(smith)} smith numbers below #{m}:" IO.puts "First 10: #{Enum.take(smith,10) |> Enum.join(", ")}" IO.puts "Last 10: #{Enum.take(smith,-10) |> Enum.join(", ")}"</lang>
- Output:
376 smith numbers below 10000: First 10: 4, 22, 27, 58, 85, 94, 121, 166, 202, 265 Last 10: 9843, 9849, 9861, 9880, 9895, 9924, 9942, 9968, 9975, 9985
J
Implementation: <lang J>digits=: 10&#.inv sumdig=: +/@,@digits notprime=: -.@(1&p:) smith=: #~ notprime * (=&sumdig q:)every</lang>
Task example: <lang J> #smith }.i.10000 376
q:376
2 2 2 47
8 47$smith }.i.10000 4 22 27 58 85 94 121 166 202 265 274 319 346 355 378 382 391 438 454 483 517 526 535 562 576 588 627 634 636 645 648 654 663 666 690 706 728 729 762 778 825 852 861 895 913 915 922 958 985 1086 1111 1165 1219 1255 1282 1284 1376 1449 1507 1581 1626 1633 1642 1678 1736 1755 1776 1795 1822 1842 1858 1872 1881 1894 1903 1908 1921 1935 1952 1962 1966 2038 2067 2079 2155 2173 2182 2218 2227 2265 2286 2326 2362 2366
2373 2409 2434 2461 2475 2484 2515 2556 2576 2578 2583 2605 2614 2679 2688 2722 2745 2751 2785 2839 2888 2902 2911 2934 2944 2958 2964 2965 2970 2974 3046 3091 3138 3168 3174 3226 3246 3258 3294 3345 3366 3390 3442 3505 3564 3595 3615 3622 3649 3663 3690 3694 3802 3852 3864 3865 3930 3946 3973 4054 4126 4162 4173 4185 4189 4191 4198 4209 4279 4306 4369 4414 4428 4464 4472 4557 4592 4594 4702 4743 4765 4788 4794 4832 4855 4880 4918 4954 4959 4960 4974 4981 5062 5071 5088 5098 5172 5242 5248 5253 5269 5298 5305 5386 5388 5397 5422 5458 5485 5526 5539 5602 5638 5642 5674 5772 5818 5854 5874 5915 5926 5935 5936 5946 5998 6036 6054 6084 6096 6115 6171 6178 6187 6188 6252 6259 6295 6315 6344 6385 6439 6457 6502 6531 6567 6583 6585 6603 6684 6693 6702 6718 6760 6816 6835 6855 6880 6934 6981 7026 7051 7062 7068 7078 7089 7119 7136 7186 7195 7227 7249 7287 7339 7402 7438 7447 7465 7503 7627 7674 7683 7695 7712 7726 7762 7764 7782 7784 7809 7824 7834 7915 7952 7978 8005 8014 8023 8073 8077 8095 8149 8154 8158 8185 8196 8253 8257 8277 8307 8347 8372 8412 8421 8466 8518 8545 8568 8628 8653 8680 8736 8754 8766 8790 8792 8851 8864 8874 8883 8901 8914 9015 9031 9036 9094 9166 9184 9193 9229 9274 9276 9285 9294 9296 9301 9330 9346 9355 9382 9386 9387 9396 9414 9427 9483 9522 9535 9571 9598 9633 9634 9639 9648 9657 9684 9708 9717 9735 9742 9760 9778 9840 9843 9849 9861 9880 9895 9924 9942 9968 9975 9985</lang>
(first we count how many smith numbers are in our result, then we look at the prime factors of that count - turns out that 8 rows of 47 numbers each is perfect for this task.)
Java
<lang java>import java.util.*;
public class SmithNumbers {
public static void main(String[] args) {
for (int n = 1; n < 10_000; n++) {
List<Integer> factors = primeFactors(n);
if (factors.size() > 1) {
int sum = sumDigits(n);
for (int f : factors)
sum -= sumDigits(f);
if (sum == 0)
System.out.println(n);
}
}
}
static List<Integer> primeFactors(int n) {
List<Integer> result = new ArrayList<>();
for (int i = 2; n % i == 0; n /= i)
result.add(i);
for (int i = 3; i * i <= n; i += 2) {
while (n % i == 0) {
result.add(i);
n /= i;
}
}
if (n != 1)
result.add(n);
return result; }
static int sumDigits(int n) {
int sum = 0;
while (n > 0) {
sum += (n % 10);
n /= 10;
}
return sum;
}
}</lang>
4 22 27 58 85 94 121 ... 9924 9942 9968 9975 9985
Lua
Slightly long-winded prime factor function but it's a bit faster than the 'easy' way. <lang Lua>-- Returns a boolean indicating whether n is prime function isPrime (n)
if n < 2 then return false end
if n < 4 then return true end
if n % 2 == 0 then return false end
for d = 3, math.sqrt(n), 2 do
if n % d == 0 then return false end
end
return true
end
-- Returns a table of the prime factors of n function primeFactors (n)
local pfacs, divisor = {}, 1
if n < 1 then return pfacs end
while not isPrime(n) do
while not isPrime(divisor) do divisor = divisor + 1 end
while n % divisor == 0 do
n = n / divisor
table.insert(pfacs, divisor)
end
divisor = divisor + 1
if n == 1 then return pfacs end
end
table.insert(pfacs, n)
return pfacs
end
-- Returns the sum of the digits of n function sumDigits (n)
local sum, nStr = 0, tostring(n)
for digit = 1, nStr:len() do
sum = sum + tonumber(nStr:sub(digit, digit))
end
return sum
end
-- Returns a boolean indicating whether n is a Smith number function isSmith (n)
if isPrime(n) then return false end
local sumFacs = 0
for _, v in ipairs(primeFactors(n)) do
sumFacs = sumFacs + sumDigits(v)
end
return sumFacs == sumDigits(n)
end
-- Main procedure for n = 1, 10000 do
if isSmith(n) then io.write(n .. "\t") end
end</lang> Seems silly to paste in all 376 numbers but rest assured the output agrees with https://oeis.org/A006753
Pascal
Using a segmented sieve of erathostenes and mark every number with the index of its prime factor <= sqrt(number). I use a presieved segment to reduce the time for small primes. I thought, it would be a small speed improvement ;-)
the function IncDgtSum delivers the next sum of digits very fast (2.6 s for 1 to 1e9 ) <lang pascal>program SmithNum; {$IFDEF FPC}
{$MODE objFPC} //result and useful for x64
{$CODEALIGN PROC=64}
{$ENDIF} uses
sysutils;
type
tdigit = byte; tSum = LongInt;
const
base = 10; //maxDigitCnt *(base-1) <= High(tSum) //maxDigitCnt <= High(tSum) DIV (base-1); maxDigitCnt = 16;
StartPrimNo = 6; csegsieveSIze = 2*3*5*7*11*13;//prime 0..5
type
tDgtSum = record
dgtNum : LongInt;
dgtSum : tSum;
dgts : array[0..maxDigitCnt-1] of tdigit;
end;
tNumFactype = word;
tnumFactor = record
numfacCnt: tNumFactype;
numfacts : array[1..15] of tNumFactype;
end;
tpnumFactor= ^tnumFactor;
tsieveprim = record
spPrim : Word;
spDgtsum : Word;
spOffset : LongWord;
end;
tpsieveprim = ^tsieveprim;
tsievePrimarr = array[0..6542-1] of tsieveprim; tsegmSieve = array[1..csegsieveSIze] of tnumFactor;
var
Primarr:tsievePrimarr; copySieve, actSieve : tsegmSieve; PrimDgtSum :tDgtSum; PrimCnt : NativeInt;
function IncDgtSum(var ds:tDgtSum):boolean; //add 1 to dgts and corrects sum of Digits //return if overflow happens var
i : NativeInt;
Begin
i := High(ds.dgts);
inc(ds.dgtNum);
repeat
IF ds.dgts[i] < Base-1 then
//add one and done
Begin
inc(ds.dgts[i]);
inc(ds.dgtSum);
BREAK;
end
else
Begin
ds.dgts[i] := 0;
dec(ds.dgtSum,Base-1);
end;
dec(i);
until i < Low(ds.dgts);
result := i < Low(ds.dgts)
end;
procedure OutDgtSum(const ds:tDgtSum); var
i : NativeInt;
Begin
i := Low(ds.dgts);
repeat
write(ds.dgts[i]:3);
inc(i);
until i > High(ds.dgts);
writeln(' sum of digits : ',ds.dgtSum:3);
end;
procedure OutSieve(var s:tsegmSieve); var
i,j : NativeInt;
Begin
For i := Low(s) to High(s) do
with s[i] do
Begin
write(i:6,numfacCnt:4);
For j := 1 to numfacCnt do
write(numFacts[j]:5);
writeln;
end;
end;
procedure SieveForPrimes; // sieve for all primes < High(Word) var
sieve : array of byte; pS : pByte; p,i : NativeInt;
Begin
setlength(sieve,High(Word));
Fillchar(sieve[Low(sieve)],length(sieve),#0);
pS:= @sieve[0]; //zero based
dec(pS);// make it one based
//sieve
p := 2;
repeat
i := p*p;
IF i> High(Word) then
BREAK;
repeat pS[i] := 1; inc(i,p); until i > High(Word);
repeat inc(p) until pS[p] = 0;
until false;
//now fill array of primes
fillchar(PrimDgtSum,SizeOf(PrimDgtSum),#0);
IncDgtSum(PrimDgtSum);//1
i := 0;
For p := 2 to High(Word) do
Begin
IncDgtSum(PrimDgtSum);
if pS[p] = 0 then
Begin
with PrimArr[i] do
Begin
spOffset := 2*p;//start at 2*prime
spPrim := p;
spDgtsum := PrimDgtSum.dgtSum;
end;
inc(i);
end;
end;
PrimCnt := i-1;
end;
procedure MarkWithPrime(SpIdx:NativeInt;var sf:tsegmSieve); var
i : NativeInt; pSf :^tnumFactor; MarkPrime : NativeInt;
Begin
with Primarr[SpIdx] do
Begin
MarkPrime := spPrim;
i := spOffSet;
IF i <= csegsieveSize then
Begin
pSf := @sf[i];
repeat
pSf^.numFacts[pSf^.numfacCnt+1] := SpIdx;
inc(pSf^.numfacCnt);
inc(pSf,MarkPrime);
inc(i,MarkPrime);
until i > csegsieveSize;
end;
spOffset := i-csegsieveSize;
end;
end;
procedure InitcopySieve(var cs:tsegmSieve); var
pr: NativeInt;
Begin
fillchar(cs[Low(cs)],sizeOf(cs),#0);
For Pr := 0 to 5 do
Begin
with Primarr[pr] do
spOffset := spPrim;//mark the prime too
MarkWithPrime(pr,cs);
end;
end;
procedure MarkNextSieve(var s:tsegmSieve); var
idx: NativeInt;
Begin
s:= copySieve; For idx := StartPrimNo to PrimCnt do MarkWithPrime(idx,s);
end;
function DgtSumInt(n: NativeUInt):NativeUInt; var
r : NativeUInt;
Begin
result := 0; repeat r := n div base; inc(result,n-base*r); n := r until r = 0;
end;
{function DgtSumOfFac(pN: tpnumFactor;dgtNo:tDgtSum):boolean;} function TestSmithNum(pN: tpnumFactor;dgtNo:tDgtSum):boolean; var
i,k,r,dgtSumI,dgtSumTarget : NativeUInt; pSp:tpsieveprim; pNumFact : ^tNumFactype;
Begin
i := dgtNo.dgtNum; dgtSumTarget :=dgtNo.dgtSum;
dgtSumI := 0; with pN^ do Begin k := numfacCnt; pNumFact := @numfacts[k]; end;
For k := k-1 downto 0 do
Begin
pSp := @PrimArr[pNumFact^];
r := i DIV pSp^.spPrim;
repeat
i := r;
r := r DIV pSp^.spPrim;
inc(dgtSumI,pSp^.spDgtsum);
until (i - r* pSp^.spPrim) <> 0;
IF dgtSumI > dgtSumTarget then
Begin
result := false;
EXIT;
end;
dec(pNumFact);
end;
If i <> 1 then
inc(dgtSumI,DgtSumInt(i));
result := dgtSumI = dgtSumTarget
end;
function CheckSmithNo(var s:tsegmSieve;var dgtNo:tDgtSum;Lmt:NativeInt=csegsieveSIze):NativeUInt; var
pNumFac : tpNumFactor; i : NativeInt;
Begin
result := 0;
i := low(s);
pNumFac := @s[i];
For i := i to lmt do
Begin
incDgtSum(dgtNo);
IF pNumFac^.numfacCnt<> 0 then
IF TestSmithNum(pNumFac,dgtNo) then
Begin
inc(result);
//Mark as smith number
inc(pNumFac^.numfacCnt,1 shl 15);
end;
inc(pNumFac);
end;
end;
const
limit = 100*1000*1000;
var
actualNo :tDgtSum; i,s : NativeInt;
Begin
SieveForPrimes; InitcopySieve(copySieve); i := 1; s:= -6;//- 2,3,5,7,11,13
fillchar(actualNo,SizeOf(actualNo),#0); while i < Limit-csegsieveSize do Begin MarkNextSieve(actSieve); inc(s,CheckSmithNo(actSieve,actualNo)); inc(i, csegsieveSize); end; //check the rest MarkNextSieve(actSieve); inc(s,CheckSmithNo(actSieve,actualNo,Limit-i+1)); write(s:8,' smith-numbers up to ',actualNo.dgtnum:10);
end.
</lang>
- Output:
64-Bit FPC 3.1.1 -O3 -Xs i4330 3.5 Ghz
6 smith-numbers up to 100
49 smith-numbers up to 1000
376 smith-numbers up to 10000
3294 smith-numbers up to 100000
29928 smith-numbers up to 1000000 real 0m00.064s
278411 smith-numbers up to 10000000 real 0m00.661s
2632758 smith-numbers up to 100000000 real 0m06.981s
25154060 smith-numbers up to 1000000000 real 1m14.077s
Number of Smith numbers below 10^n. 1
1:1, 2:6, 3:49, 4:376, 5:3294, 6:29928, 7:278411, 8:2632758,
9:25154060, 10:241882509, 11:2335807857, 12:22635291815,13:219935518608
Perl
<lang perl>use ntheory qw/:all/; my @smith; forcomposites {
push @smith, $_ if sumdigits($_) == sumdigits(join("",factors($_)));
} 10000-1; say scalar(@smith), " Smith numbers below 10000."; say "@smith";</lang>
- Output:
376 Smith numbers below 10000. 4 22 27 58 85 94 121 166 202 ... 9924 9942 9968 9975 9985
Perl 6
<lang perl6>constant @primes = 2, |(3, 5, 7 ... *).grep: *.is-prime;
multi factors ( 1 ) { 1 } multi factors ( Int $remainder is copy ) {
gather for @primes -> $factor {
# if remainder < factor², we're done
if $factor * $factor > $remainder {
take $remainder if $remainder > 1;
last;
}
# How many times can we divide by this prime?
while $remainder %% $factor {
take $factor;
last if ($remainder div= $factor) === 1;
}
}
}
- Code above here is verbatim from RC:Count_in_factors#Perl6
sub is_smith_number ( Int $n ) {
(!$n.is-prime) and ( [+] $n.comb ) == ( [+] factors($n).join.comb );
}
my @s = grep &is_smith_number, 2 ..^ 10_000; say "{@s.elems} Smith numbers below 10_000"; say 'First 10: ', @s[ ^10 ]; say 'Last 10: ', @s[ *-10 .. * ];</lang>
- Output:
376 Smith numbers below 10_000 First 10: (4 22 27 58 85 94 121 166 202 265) Last 10: (9843 9849 9861 9880 9895 9924 9942 9968 9975 9985)
REXX
unoptimized
<lang rexx>/*REXX program finds (and maybe displays) Smith (or joke) numbers up to a given N.*/ parse arg N . /*obtain optional argument from the CL.*/ if N== | N=="," then N=10000 /*Not specified? Then use the default.*/ tell= (N>0); N=abs(N) /*use the │N│ for computing (below).*/ w=length(N) /*W: used for aligning Smith numbers. */
- =0 /*#: Smith numbers found (so far). */
@=; do j=4 to N; /*process almost all numbers up to N. */
if sumD(j) \== sumfactr(j) then iterate /*Not a Smith number? Then ignore it.*/
#=#+1 /*bump the Smith number counter. */
if \tell then iterate /*Not showing the numbers? Keep looking*/
@=@ right(j, w); if length(@)>75 then do; say substr(@, 2); @=; end
end /*j*/ /* [↑] if N>0, then display Smith #s.*/
if @\== then say substr(@, 2) /*if any residual Smith #s, display 'em*/ say /* [↓] display the number of Smith #s.*/ say # ' Smith numbers found ≤ ' N"." /*display number of Smith numbers found*/ exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ sumD: parse arg x 1 s 2; do d=2 for length(x)-1; s=s+substr(x,d,1); end; return s /*──────────────────────────────────────────────────────────────────────────────────────*/ sumFactr: procedure; parse arg z 1 n; $=0; f=0 /*Z and N are both arg1*/
do while z//2==0; $=$+2; f=f+1; z=z% 2; end /*maybe add factor of 2*/
do while z//3==0; $=$+3; f=f+1; z=z% 3; end /* " " " " 3*/
/* ___*/
do j=5 by 2 while j<=z & j*j<=n /*minimum of Z or √ N */
if j//3==0 then iterate /*skip factors that ÷ 3*/
do while z//j==0; f=f+1; $=$+sumD(j); z=z%j; end /*maybe reduce Z by J */
end /*j*/ /* [↓] Z: what's left*/
if z\==1 then do; f=f+1; $=$+sumD(z); end /*Residual? Then add Z*/
if f<2 then return 0 /*Prime? Not a Smith#*/
return $ /*else return sum digs.*/</lang>
output when using the default number:
4 22 27 58 85 94 121 166 202 265 274 319 346 355 378 382 391 438 454 483 517 526 535 562 576 588 627 634 636 645 648 654 663 666 690 706 728 729 762 778 825 852 861 895 913 915 922 958 985 1086 1111 1165 1219 1255 1282 1284 1376 1449 1507 1581 1626 1633 1642 1678 1736 1755 1776 1795 1822 1842 1858 1872 1881 1894 1903 1908 1921 1935 1952 1962 1966 2038 2067 2079 2155 2173 2182 2218 2227 2265 2286 2326 2362 2366 2373 2409 2434 2461 2475 2484 2515 2556 2576 2578 2583 2605 2614 2679 2688 2722 2745 2751 2785 2839 2888 2902 2911 2934 2944 2958 2964 2965 2970 2974 3046 3091 3138 3168 3174 3226 3246 3258 3294 3345 3366 3390 3442 3505 3564 3595 3615 3622 3649 3663 3690 3694 3802 3852 3864 3865 3930 3946 3973 4054 4126 4162 4173 4185 4189 4191 4198 4209 4279 4306 4369 4414 4428 4464 4472 4557 4592 4594 4702 4743 4765 4788 4794 4832 4855 4880 4918 4954 4959 4960 4974 4981 5062 5071 5088 5098 5172 5242 5248 5253 5269 5298 5305 5386 5388 5397 5422 5458 5485 5526 5539 5602 5638 5642 5674 5772 5818 5854 5874 5915 5926 5935 5936 5946 5998 6036 6054 6084 6096 6115 6171 6178 6187 6188 6252 6259 6295 6315 6344 6385 6439 6457 6502 6531 6567 6583 6585 6603 6684 6693 6702 6718 6760 6816 6835 6855 6880 6934 6981 7026 7051 7062 7068 7078 7089 7119 7136 7186 7195 7227 7249 7287 7339 7402 7438 7447 7465 7503 7627 7674 7683 7695 7712 7726 7762 7764 7782 7784 7809 7824 7834 7915 7952 7978 8005 8014 8023 8073 8077 8095 8149 8154 8158 8185 8196 8253 8257 8277 8307 8347 8372 8412 8421 8466 8518 8545 8568 8628 8653 8680 8736 8754 8766 8790 8792 8851 8864 8874 8883 8901 8914 9015 9031 9036 9094 9166 9184 9193 9229 9274 9276 9285 9294 9296 9301 9330 9346 9355 9382 9386 9387 9396 9414 9427 9483 9522 9535 9571 9598 9633 9634 9639 9648 9657 9684 9708 9717 9735 9742 9760 9778 9840 9843 9849 9861 9880 9895 9924 9942 9968 9975 9985 376 Smith numbers found ≤ 10000.
optimized
This REXX version uses a faster version of the sumFactr function; it's about
three times faster than the unoptimized version using a (negative) one million.
<lang rexx>/*REXX program finds (and maybe displays) Smith (or joke) numbers up to a given N.*/
parse arg N . /*obtain optional argument from the CL.*/
if N== | N=="," then N=10000 /*Not specified? Then use the default.*/
tell= (N>0); N=abs(N) /*use the │N│ for computing (below).*/
- =0 /*the number of Smith numbers (so far).*/
w=length(N) /*W: used for aligning Smith numbers. */ @=; do j=4 to N /*process almost all numbers up to N. */
if sumD(j) \== sumfactr(j) then iterate /*Not a Smith number? Then ignore it.*/
#=#+1 /*bump the Smith number counter. */
if \tell then iterate /*Not showing the numbers? Keep looking*/
@=@ right(j, w); if length(@)>75 then do; say substr(@, 2); @=; end
end /*j*/ /* [↑] if N>0, then display Smith #s.*/
if @\== then say substr(@, 2) /*if any residual Smith #s, display 'em*/ say /* [↓] display the number of Smith #s.*/ say # ' Smith numbers found ≤ ' N"." /*display number of Smith numbers found*/ exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ sumD: parse arg x 1 s 2; do d=2 for length(x)-1; s=s+substr(x,d,1); end; return s /*──────────────────────────────────────────────────────────────────────────────────────*/ sumFactr: procedure expose !.; parse arg z 1 n; $=0; f=0 /*Z and N are both arg1*/
do while z//2==0; $=$+2; f=f+1; z=z% 2; end /*maybe add factor of 2*/
do while z//3==0; $=$+3; f=f+1; z=z% 3; end /* " " " " 3*/
do while z//5==0; $=$+5; f=f+1; z=z% 5; end /* " " " " 5*/
do while z//7==0; $=$+7; f=f+1; z=z% 7; end /* " " " " 7*/
t=z; r=0; q=1; do while q<=t; q=q*4; end /*R: will be iSqrt(Z).*/
do while q>1; q=q%4; _=t-r-q; r=r%2; if _>=0 then do; t=_; r=r+q; end
end /*while q>1*/ /* [↑] compute int. SQRT(Z)*/
do j=11 by 6 to r while j<=z /*skip factors that are ÷ 3*/
parse var j -1 _; if _\==5 then, /*is last dec. digit ¬a 5 ?*/
do while z//j==0; f=f+1; $=$+sumD(j); z=z%j; end /*maybe reduce Z by J*/
if _==3 then iterate; y=j+2
do while z//y==0; f=f+1; $=$+sumD(y); z=z%y; end /*maybe reduce Z by Y*/
end /*j*/ /* [↓] Z is what's left. */
if z\==1 then do; f=f+1; $=$+sumD(z); end /*if a residual, then add Z*/
if f<2 then return 0 /*Is prime? N not Smith #.*/
return $ /*else, return sum of digs.*/</lang>
output when using the input of (negative) one million: -1000000
29928 Smith numbers found ≤ 1000000.
Ruby
<lang ruby>require "prime"
class Fixnum
def sum_digits to_s.chars.map(&:to_i).inject(:+) end
def smith?
return false if prime?
sum_digits == prime_division.map{|pr,n| pr.sum_digits * n}.inject(:+)
end
end
n = 10_000 res = 1.upto(n).select(&:smith?)
puts "#{res.size} smith numbers below #{n}:
- {res.first(5).join(", ")},... #{res.last(5).join(", ")}"</lang>
- Output:
376 smith numbers below 10000: 4, 22, 27, 58, 85,... 9924, 9942, 9968, 9975, 9985
Sidef
<lang ruby>var primes = Enumerator({ |callback|
static primes = Hash()
var p = 2
loop {
callback(p)
p = (primes{p} := p.next_prime)
}
})
func factors(remainder) {
remainder == 1 && return([remainder])
gather {
primes.each { |factor|
if (factor*factor > remainder) {
take(remainder) if (remainder > 1)
break
}
while (factor.divides(remainder)) {
take(factor)
break if ((remainder /= factor) == 1)
}
}
}
}
func is_smith_number(n) {
!n.is_prime && (n.digits.sum == factors(n).join.to_i.digits.sum)
}
var s = range(2, 10_000).grep { is_smith_number(_) } say "#{s.len} Smith numbers below 10_000" say "First 10: #{s.first(10)}" say "Last 10: #{s.last(10)}"</lang>
- Output:
376 Smith numbers below 10_000 First 10: [4, 22, 27, 58, 85, 94, 121, 166, 202, 265] Last 10: [9843, 9849, 9861, 9880, 9895, 9924, 9942, 9968, 9975, 9985]
zkl
Uses the code (primeFactors) from Prime decomposition#zkl. <lang zkl>fcn smithNumbers(N=0d10_000){ // -->(Smith numbers to N)
[2..N].filter(fcn(n){
(pfs:=primeFactors(n)).len()>1 and
n.split().sum(0)==primeFactors(n).apply("split").flatten().sum(0)
})
}</lang> <lang zkl>sns:=smithNumbers(); sns.toString(*).println(" ",sns.len()," numbers");</lang>
- Output:
L(4,22,27,58,85,94,121,166,202,265,274,319,346,355,378,382,391, ... 3091,3138,3168,3174,3226,3246,3258,3294,3345,3366,3390,3442,3505, ... 9942,9968,9975,9985) 376 numbers