Factorions
A factorion is a natural number that equals the sum of the factorials of its digits.
- Definition
- Example
145 is a factorion in base 10 because:
1! + 4! + 5! = 1 + 24 + 120 = 145
It can be shown (see the Wikipedia article below) that no factorion in base 10 can exceed 1,499,999.
- Task
Write a program in your language to demonstrate, by calculating and printing out the factorions, that shows:
- There are 3 factorions in base 9
- There are 4 factorions in base 10
- There are 5 factorions in base 11
- There are 2 factorions in base 12 (up to the same upper bound as for base 10)
ALGOL 68
<lang algol68>BEGIN
# cache factorials from 0 to 11 #
[ 0 : 11 ]INT fact;
fact[0] := 1;
FOR n TO 11 DO
fact[n] := fact[n-1] * n
OD;
FOR b FROM 9 TO 12 DO
print( ( "The factorions for base ", whole( b, 0 ), " are:", newline ) );
FOR i TO 1500000 - 1 DO
INT sum := 0;
INT j := i;
WHILE j > 0 DO
sum +:= fact[ j MOD b ];
j OVERAB b
OD;
IF sum = i THEN print( ( whole( i, 0 ), " " ) ) FI
OD;
print( ( newline ) )
OD
END</lang>
- Output:
The factorions for base 9 are: 1 2 41282 The factorions for base 10 are: 1 2 145 40585 The factorions for base 11 are: 1 2 26 48 40472 The factorions for base 12 are: 1 2
AWK
<lang AWK>
- syntax: GAWK -f FACTORIONS.AWK
- converted from C
BEGIN {
fact[0] = 1 # cache factorials from 0 to 11
for (n=1; n<12; ++n) {
fact[n] = fact[n-1] * n
}
for (b=9; b<=12; ++b) {
printf("base %d factorions:",b)
for (i=1; i<1500000; ++i) {
sum = 0
j = i
while (j > 0) {
d = j % b
sum += fact[d]
j = int(j/b)
}
if (sum == i) {
printf(" %d",i)
}
}
printf("\n")
}
exit(0)
} </lang>
- Output:
base 9 factorions: 1 2 41282 base 10 factorions: 1 2 145 40585 base 11 factorions: 1 2 26 48 40472 base 12 factorions: 1 2
C
<lang c>#include <stdio.h>
int main() {
int n, b, d;
unsigned long long i, j, sum, fact[12];
// cache factorials from 0 to 11
fact[0] = 1;
for (n = 1; n < 12; ++n) {
fact[n] = fact[n-1] * n;
}
for (b = 9; b <= 12; ++b) {
printf("The factorions for base %d are:\n", b);
for (i = 1; i < 1500000; ++i) {
sum = 0;
j = i;
while (j > 0) {
d = j % b;
sum += fact[d];
j /= b;
}
if (sum == i) printf("%llu ", i);
}
printf("\n\n");
}
return 0;
}</lang>
- Output:
The factorions for base 9 are: 1 2 41282 The factorions for base 10 are: 1 2 145 40585 The factorions for base 11 are: 1 2 26 48 40472 The factorions for base 12 are: 1 2
Factor
<lang factor>USING: formatting io kernel math math.parser math.ranges memoize prettyprint sequences ; IN: rosetta-code.factorions
! Memoize factorial function MEMO: factorial ( n -- n! ) [ 1 ] [ [1,b] product ] if-zero ;
- factorion? ( n base -- ? )
dupd >base string>digits [ factorial ] map-sum = ;
- show-factorions ( limit base -- )
dup "The factorions for base %d are:\n" printf [ [1,b) ] dip [ dupd factorion? [ pprint bl ] [ drop ] if ] curry each nl ;
1,500,000 9 12 [a,b] [ show-factorions nl ] with each</lang>
- Output:
The factorions for base 9 are: 1 2 41282 The factorions for base 10 are: 1 2 145 40585 The factorions for base 11 are: 1 2 26 48 40472 The factorions for base 12 are: 1 2
Fōrmulæ
In this page you can see the solution of this task.
Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text (more info). Moreover, there can be multiple visual representations of the same program. Even though it is possible to have textual representation —i.e. XML, JSON— they are intended for transportation effects more than visualization and edition.
The option to show Fōrmulæ programs and their results is showing images. Unfortunately images cannot be uploaded in Rosetta Code.
Go
<lang go>package main
import (
"fmt" "strconv"
)
func main() {
// cache factorials from 0 to 11
var fact [12]uint64
fact[0] = 1
for n := uint64(1); n < 12; n++ {
fact[n] = fact[n-1] * n
}
for b := 9; b <= 12; b++ {
fmt.Printf("The factorions for base %d are:\n", b)
for i := uint64(1); i < 1500000; i++ {
digits := strconv.FormatUint(i, b)
sum := uint64(0)
for _, digit := range digits {
if digit < 'a' {
sum += fact[digit-'0']
} else {
sum += fact[digit+10-'a']
}
}
if sum == i {
fmt.Printf("%d ", i)
}
}
fmt.Println("\n")
}
}</lang>
- Output:
The factorions for base 9 are: 1 2 41282 The factorions for base 10 are: 1 2 145 40585 The factorions for base 11 are: 1 2 26 48 40472 The factorions for base 12 are: 1 2
Julia
<lang julia>isfactorian(n, base) = mapreduce(factorial, +, map(c -> parse(Int, c, base=16), split(string(n, base=base), ""))) == n
printallfactorian(base) = println("Factorians for base $base: ", [n for n in 1:100000 if isfactorian(n, base)])
foreach(printallfactorian, 9:12)
</lang>
- Output:
Factorians for base 9: [1, 2, 41282] Factorians for base 10: [1, 2, 145, 40585] Factorians for base 11: [1, 2, 26, 48, 40472] Factorians for base 12: [1, 2]
OCaml
<lang ocaml>let () =
(* cache factorials from 0 to 11 *) let fact = Array.make 12 0 in fact.(0) <- 1; for n = 1 to pred 12 do fact.(n) <- fact.(n-1) * n; done;
for b = 9 to 12 do
Printf.printf "The factorions for base %d are:\n" b;
for i = 1 to pred 1_500_000 do
let sum = ref 0 in
let j = ref i in
while !j > 0 do
let d = !j mod b in
sum := !sum + fact.(d);
j := !j / b;
done;
if !sum = i then (print_int i; print_string " ")
done;
print_string "\n\n";
done</lang>
Perl
<lang perl>use strict; use warnings; use ntheory qw/factorial todigits/;
my $limit = 1500000;
for my $b (9 .. 12) {
print "Factorions in base $b:\n"; $_ == factorial($_) and print "$_ " for 0..$b-1;
for my $i (1 .. int $limit/$b) {
my $sum;
my $prod = $i * $b;
for (reverse todigits($i, $b)) {
$sum += factorial($_);
$sum = 0 && last if $sum > $prod;
}
next if $sum == 0;
($sum + factorial($_) == $prod + $_) and print $prod+$_ . ' ' for 0..$b-1;
}
print "\n\n";
}</lang>
- Output:
Factorions in base 9: 1 2 41282 Factorions in base 10: 1 2 145 40585 Factorions in base 11: 1 2 26 48 40472 Factorions in base 12: 1 2
Alternatively, a more efficient approach:
<lang perl>use 5.020; use ntheory qw(:all); use experimental qw(signatures); use Algorithm::Combinatorics qw(combinations_with_repetition);
sub max_power ($base = 10) {
my $m = 1;
my $f = factorial($base - 1);
while ($m * $f >= $base**($m-1)) {
$m += 1;
}
return $m-1;
}
sub factorions ($base = 10) {
my @result;
my @digits = (0 .. $base-1);
my @factorial = map { factorial($_) } @digits;
foreach my $k (1 .. max_power($base)) {
my $iter = combinations_with_repetition(\@digits, $k);
while (my $comb = $iter->next) {
my $n = vecsum(map { $factorial[$_] } @$comb);
if (join(' ', sort { $a <=> $b } todigits($n, $base)) eq join(' ', @$comb)) {
push @result, $n;
}
}
}
return @result;
}
foreach my $base (2 .. 14) {
my @r = factorions($base); say "Factorions in base $base are (@r)";
}</lang>
- Output:
Factorions in base 2 are (1 2) Factorions in base 3 are (1 2) Factorions in base 4 are (1 2 7) Factorions in base 5 are (1 2 49) Factorions in base 6 are (1 2 25 26) Factorions in base 7 are (1 2) Factorions in base 8 are (1 2) Factorions in base 9 are (1 2 41282) Factorions in base 10 are (1 2 145 40585) Factorions in base 11 are (1 2 26 48 40472) Factorions in base 12 are (1 2) Factorions in base 13 are (1 2 519326767) Factorions in base 14 are (1 2 12973363226)
Phix
<lang Phix>-- cache factorials from 0 to 11 sequence fact = repeat(1,12) for n=2 to length(fact) do
fact[n] = fact[n-1]*(n-1)
end for
for b=9 to 12 do
printf(1,"The factorions for base %d are:\n", b)
for i=1 to 1499999 do
atom total = 0, j = i, d
while j>0 and total<=i do
d = remainder(j,b)
total += fact[d+1]
j = floor(j/b)
end while
if total==i then printf(1,"%d ", i) end if
end for
printf(1,"\n\n")
end for</lang>
- Output:
The factorions for base 9 are: 1 2 41282 The factorions for base 10 are: 1 2 145 40585 The factorions for base 11 are: 1 2 26 48 40472 The factorions for base 12 are: 1 2
Racket
<lang racket>#lang racket
(define fact
(curry list-ref (for/fold ([result (list 1)] #:result (reverse result))
([x (in-range 1 20)])
(cons (* x (first result)) result))))
(for ([b (in-range 9 13)])
(printf "The factorions for base ~a are:\n" b)
(for ([i (in-range 1 1500000)])
(let loop ([sum 0] [n i])
(cond
[(positive? n) (loop (+ sum (fact (modulo n b))) (quotient n b))]
[(= sum i) (printf "~a " i)])))
(newline))</lang>
- Output:
The factorions for base 9 are: 1 2 41282 The factorions for base 10 are: 1 2 145 40585 The factorions for base 11 are: 1 2 26 48 40472 The factorions for base 12 are: 1 2
Raku
(formerly Perl 6)
<lang perl6>constant @factorial = 1, |[\*] 1..*;
constant $limit = 1500000;
constant $bases = 9 .. 12;
my @result;
$bases.race(:1batch).map: -> $base {
@result[$base] = "\nFactorions in base $base:\n1 2";
sink (1 .. $limit div $base).map: -> $i {
my $product = $i * $base;
my $partial;
for $i.polymod($base xx *) {
$partial += @factorial[$_];
last if $partial > $product
}
next if $partial > $product;
my $sum;
for ^$base {
last if ($sum = $partial + @factorial[$_]) > $product + $_;
@result[$base] ~= " $sum" and last if $sum == $product + $_
}
}
}
.say for @result[$bases];</lang>
- Output:
Factorions in base 9: 1 2 41282 Factorions in base 10: 1 2 145 40585 Factorions in base 11: 1 2 26 48 40472 Factorions in base 12: 1 2
REXX
<lang rexx>/*REXX program calculates and displays factorions in bases nine ───► twelve. */ parse arg LOb HIb lim . /*obtain optional arguments from the CL*/ if LOb== | LOb=="," then LOb= 9 /*Not specified? Then use the default.*/ if HIb== | HIb=="," then HIb= 12 /* " " " " " " */ if lim== | lim=="," then lim= 1500000 - 1 /* " " " " " " */
do fact=0 to HIb; !.fact= !(fact) /*use memoization for factorials. */ end /*fact*/
do base=LOb to HIb /*process all the required bases. */
@= 1 2 /*initialize the list (@) to null. */
do j=3 for lim-2; $= 0 /*initialize the sum ($) to zero. */
t= j /*define the target (for the sum !'s).*/
do until t==0; d= t // base /*obtain a "digit". */
$= $ + !.d /*add !(d) to sum. */
t= t % base /*get a new target. */
end /*until*/
if $==j then @= @ j /*Good factorial sum? Then add to list.*/
end /*i*/
say
say 'The factorions for base ' right( base, length(HIb) ) " are: " @
end /*base*/
exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ !: procedure; arg x; !=1; do j=2 to x; !=!*j; end; return ! /*calc factorials.*/</lang>
- output when using the default inputs:
The factorions for base 9 are: 1 2 41282 The factorions for base 10 are: 1 2 145 40585 The factorions for base 11 are: 1 2 26 48 40472 The factorions for base 12 are: 1 2
Sidef
<lang ruby>func max_power(b = 10) {
var m = 1
var f = (b-1)!
while (m*f >= b**(m-1)) {
m += 1
}
return m-1
}
func factorions(b = 10) {
var result = []
var digits = @^b
var fact = digits.map { _! }
for k in (1 .. max_power(b)) {
digits.combinations_with_repetition(k, {|*comb|
var n = comb.sum_by { fact[_] }
if (n.digits(b).sort == comb) {
result << n
}
})
}
return result
}
for b in (2..12) {
var r = factorions(b)
say "Base #{'%2d' % b} factorions: #{r}"
}</lang>
- Output:
Base 2 factorions: [1, 2] Base 3 factorions: [1, 2] Base 4 factorions: [1, 2, 7] Base 5 factorions: [1, 2, 49] Base 6 factorions: [1, 2, 25, 26] Base 7 factorions: [1, 2] Base 8 factorions: [1, 2] Base 9 factorions: [1, 2, 41282] Base 10 factorions: [1, 2, 145, 40585] Base 11 factorions: [1, 2, 26, 48, 40472] Base 12 factorions: [1, 2]
Swift
<lang swift>var fact = Array(repeating: 0, count: 12)
fact[0] = 1
for n in 1..<12 {
fact[n] = fact[n - 1] * n
}
for b in 9...12 {
print("The factorions for base \(b) are:")
for i in 1..<1500000 {
var sum = 0
var j = i
while j > 0 {
sum += fact[j % b]
j /= b
}
if sum == i {
print("\(i)", terminator: " ")
fflush(stdout)
}
}
print("\n")
}</lang>
- Output:
The factorions for base 9 are: 1 2 41282 The factorions for base 10 are: 1 2 145 40585 The factorions for base 11 are: 1 2 26 48 40472 The factorions for base 12 are: 1 2
zkl
<lang zkl>var facts=[0..12].pump(List,fcn(n){ (1).reduce(n,fcn(N,n){ N*n },1) }); #(1,1,2,6....) fcn factorions(base){
fs:=List();
foreach n in ([1..1_499_999]){
sum,j := 0,n;
while(j){
sum+=facts[j%base]; j/=base;
}
if(sum==n) fs.append(n);
}
fs
}</lang> <lang zkl>foreach n in ([9..12]){
println("The factorions for base %2d are: ".fmt(n),factorions(n).concat(" "));
}</lang>
- Output:
The factorions for base 9 are: 1 2 41282 The factorions for base 10 are: 1 2 145 40585 The factorions for base 11 are: 1 2 26 48 40472 The factorions for base 12 are: 1 2