Weird numbers
From Rosetta Code
You are encouraged to solve this task according to the task description, using any language you may know.
In number theory, a weird number is a natural number that is abundant but not semiperfect.
In other words, the sum of the proper divisors (divisors including 1 but not itself) of the number is greater than the number, but no subset of those divisors sums to the number itself.
E.G. 12 is not a weird number. It is abundant; the proper divisors 1, 2, 3, 4 & 6 sum to 16, but it is semiperfect, 6 + 4 + 2 == 12.
70 is a weird number. It is abundant; the proper divisors 1, 2, 5, 7, 10, 14 & 35 sum to 74, but there is no subset of proper divisors that sum to 70.
- Task
Find and display, here on this page, the first 25 weird numbers.
- See also
Contents
AppleScript[edit]
Applescript is not the recommended apparatus for this kind of experiment.
(Though after about 6 seconds (on this system) it does yield the first 25, and intermediates can be logged in the Messages channel of macOS Script Editor).
on run
take(25, weirds())
-- Gets there, but takes about 6 seconds on this system,
-- (logging intermediates through the Messages channel, for the impatient :-)
end run
-- weirds :: Gen [Int]
on weirds()
script
property x : 1
property v : 0
on |λ|()
repeat until isWeird(x)
set x to 1 + x
end repeat
set v to x
log v
set x to 1 + x
return v
end |λ|
end script
end weirds
-- isWeird :: Int -> Bool
on isWeird(n)
set ds to descProperDivisors(n)
set d to sum(ds) - n
0 < d and not hasSum(d, ds)
end isWeird
-- hasSum :: Int -> [Int] -> Bool
on hasSum(n, xs)
if {} ≠ xs then
set h to item 1 of xs
set t to rest of xs
if n < h then
hasSum(n, t)
else
n = h or hasSum(n - h, t) or hasSum(n, t)
end if
else
false
end if
end hasSum
-- GENERIC ------------------------------------------------
-- descProperDivisors :: Int -> [Int]
on descProperDivisors(n)
if n = 1 then
{1}
else
set realRoot to n ^ (1 / 2)
set intRoot to realRoot as integer
set blnPerfect to intRoot = realRoot
-- isFactor :: Int -> Bool
script isFactor
on |λ|(x)
n mod x = 0
end |λ|
end script
-- Factors up to square root of n,
set lows to filter(isFactor, enumFromTo(1, intRoot))
-- and cofactors of these beyond the square root,
-- integerQuotient :: Int -> Int
script integerQuotient
on |λ|(x)
(n / x) as integer
end |λ|
end script
set t to rest of lows
if blnPerfect then
set xs to t
else
set xs to lows
end if
map(integerQuotient, t) & (reverse of xs)
end if
end descProperDivisors
-- enumFromTo :: (Int, Int) -> [Int]
on enumFromTo(m, n)
if m ≤ n then
set lst to {}
repeat with i from m to n
set end of lst to i
end repeat
return lst
else
return {}
end if
end enumFromTo
-- filter :: (a -> Bool) -> [a] -> [a]
on filter(f, xs)
tell mReturn(f)
set lst to {}
set lng to length of xs
repeat with i from 1 to lng
set v to item i of xs
if |λ|(v, i, xs) then set end of lst to v
end repeat
return lst
end tell
end filter
-- foldl :: (a -> b -> a) -> a -> [b] -> a
on foldl(f, startValue, xs)
tell mReturn(f)
set v to startValue
set lng to length of xs
repeat with i from 1 to lng
set v to |λ|(v, item i of xs, i, xs)
end repeat
return v
end tell
end foldl
-- map :: (a -> b) -> [a] -> [b]
on map(f, xs)
tell mReturn(f)
set lng to length of xs
set lst to {}
repeat with i from 1 to lng
set end of lst to |λ|(item i of xs, i, xs)
end repeat
return lst
end tell
end map
-- sum :: [Num] -> Num
on sum(xs)
script add
on |λ|(a, b)
a + b
end |λ|
end script
foldl(add, 0, xs)
end sum
-- take :: Int -> Gen [a] -> [a]
on take(n, xs)
set ys to {}
repeat with i from 1 to n
set v to xs's |λ|()
if missing value is v then
return ys
else
set end of ys to v
end if
end repeat
return ys
end take
-- Lift 2nd class handler function into 1st class script wrapper
-- mReturn :: First-class m => (a -> b) -> m (a -> b)
on mReturn(f)
if script is class of f then
f
else
script
property |λ| : f
end script
end if
end mReturn
- Output:
{70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, 10990, 11410, 11690, 12110, 12530, 12670, 13370, 13510, 13790, 13930, 14770, 15610, 15890, 16030, 16310}
D[edit]
import std.algorithm;
import std.array;
import std.stdio;
int[] divisors(int n) {
int[] divs = [1];
int[] divs2;
for (int i = 2; i * i <= n; i++) {
if (n % i == 0) {
int j = n / i;
divs ~= i;
if (i != j) {
divs2 ~= j;
}
}
}
divs2 ~= divs.reverse;
return divs2;
}
bool abundant(int n, int[] divs) {
return divs.sum() > n;
}
bool semiperfect(int n, int[] divs) {
if (divs.length > 0) {
auto h = divs[0];
auto t = divs[1..$];
if (n < h) {
return semiperfect(n, t);
} else {
return n == h
|| semiperfect(n - h, t)
|| semiperfect(n, t);
}
} else {
return false;
}
}
bool[] sieve(int limit) {
// false denotes abundant and not semi-perfect.
// Only interested in even numbers >= 2
auto w = uninitializedArray!(bool[])(limit);
w[] = false;
for (int i = 2; i < limit; i += 2) {
if (w[i]) continue;
auto divs = divisors(i);
if (!abundant(i, divs)) {
w[i] = true;
} else if (semiperfect(i, divs)) {
for (int j = i; j < limit; j += i) {
w[j] = true;
}
}
}
return w;
}
void main() {
auto w = sieve(17_000);
int count = 0;
int max = 25;
writeln("The first 25 weird numbers:");
for (int n = 2; count < max; n += 2) {
if (!w[n]) {
write(n, ' ');
count++;
}
}
writeln;
}
- Output:
70 836 4030 5830 7192 7912 9272 10430 10570 10792 10990 11410 11690 12110 12530 12670 13370 13510 13790 13930 14770 15610 15890 16030 16310
F#[edit]
let divisors n = [1..n/2] |> List.filter (fun x->n % x = 0)
let abundant (n:int) divs = Seq.sum(divs) > n
let rec semiperfect (n:int) (divs:List<int>) =
if divs.Length > 0 then
let h = divs.Head
let t = divs.Tail
if n < h then
semiperfect n t
else
n = h || (semiperfect (n - h) t) || (semiperfect n t)
else false
let weird n =
let d = divisors n
if abundant n d then
not(semiperfect n d)
else
false
[<EntryPoint>]
let main _ =
let mutable i = 1
let mutable count = 0
while (count < 25) do
if (weird i) then
count <- count + 1
printf "%d -> %d\n" count i
i <- i + 1
0 // return an integer exit code
- Output:
1 -> 70 2 -> 836 3 -> 4030 4 -> 5830 5 -> 7192 6 -> 7912 7 -> 9272 8 -> 10430 9 -> 10570 10 -> 10792 11 -> 10990 12 -> 11410 13 -> 11690 14 -> 12110 15 -> 12530 16 -> 12670 17 -> 13370 18 -> 13510 19 -> 13790 20 -> 13930 21 -> 14770 22 -> 15610 23 -> 15890 24 -> 16030 25 -> 16310
Factor[edit]
The has-sum? word is a translation of the Haskell function.
USING: combinators.short-circuit io kernel lists lists.lazy
locals math math.primes.factors prettyprint sequences ;
IN: rosetta-code.weird-numbers
:: has-sum? ( n seq -- ? )
seq [ f ] [
unclip-slice :> ( xs x )
n x < [ n xs has-sum? ] [
{
[ n x = ]
[ n x - xs has-sum? ]
[ n xs has-sum? ]
} 0||
] if
] if-empty ;
: weird? ( n -- ? )
dup divisors but-last reverse
{ [ sum < ] [ has-sum? not ] } 2&& ;
: weirds ( -- list ) 1 lfrom [ weird? ] lfilter ;
: weird-numbers-demo ( -- )
"First 25 weird numbers:" print
25 weirds ltake list>array . ;
MAIN: weird-numbers-demo
- Output:
First 25 weird numbers:
{
70
836
4030
5830
7192
7912
9272
10430
10570
10792
10990
11410
11690
12110
12530
12670
13370
13510
13790
13930
14770
15610
15890
16030
16310
}
Go[edit]
This takes advantage of Hout's analysis (see talk page) when testing for primitive semi-perfect numbers.
It also uses a sieve so we can make use of the fact that all multiples of a semi-perfect number are themselves semi-perfect.
Runs in about 40 ms on a Celeron N3050 @1.6Ghz. The first fifty (with a sieve size of 27000) takes roughly double that.
When run on the same machine, the 'tweaked' version (linked to below) is about 3 times faster than this.
package main
import "fmt"
func divisors(n int) []int {
divs := []int{1}
divs2 := []int{}
for i := 2; i*i <= n; i++ {
if n%i == 0 {
j := n / i
divs = append(divs, i)
if i != j {
divs2 = append(divs2, j)
}
}
}
for i := len(divs) - 1; i >= 0; i-- {
divs2 = append(divs2, divs[i])
}
return divs2
}
func abundant(n int, divs []int) bool {
sum := 0
for _, div := range divs {
sum += div
}
return sum > n
}
func semiperfect(n int, divs []int) bool {
le := len(divs)
if le > 0 {
h := divs[0]
t := divs[1:]
if n < h {
return semiperfect(n, t)
} else {
return n == h || semiperfect(n-h, t) || semiperfect(n, t)
}
} else {
return false
}
}
func sieve(limit int) []bool {
// false denotes abundant and not semi-perfect.
// Only interested in even numbers >= 2
w := make([]bool, limit)
for i := 2; i < limit; i += 2 {
if w[i] {
continue
}
divs := divisors(i)
if !abundant(i, divs) {
w[i] = true
} else if semiperfect(i, divs) {
for j := i; j < limit; j += i {
w[j] = true
}
}
}
return w
}
func main() {
w := sieve(17000)
count := 0
const max = 25
fmt.Println("The first 25 weird numbers are:")
for n := 2; count < max; n += 2 {
if !w[n] {
fmt.Printf("%d ", n)
count++
}
}
fmt.Println()
}
- Output:
The first 25 weird numbers are: 70 836 4030 5830 7192 7912 9272 10430 10570 10792 10990 11410 11690 12110 12530 12670 13370 13510 13790 13930 14770 15610 15890 16030 16310
Tweaked[edit]
Link to a tweaked version at Try it Online!
Runs in under 6.0ms on the Tio server. The first fifty (with a sieve size of 26,533) takes under 12.0ms. Comments added where tweaks were applied
Haskell[edit]
weirds :: [Int]
weirds = filter abundantNotSemiperfect [1 ..]
abundantNotSemiperfect :: Int -> Bool
abundantNotSemiperfect n =
let ds = descProperDivisors n
d = sum ds - n
in 0 < d && not (hasSum d ds)
hasSum :: Int -> [Int] -> Bool
hasSum _ [] = False
hasSum n (x:xs)
| n < x = hasSum n xs
| otherwise = (n == x) || hasSum (n - x) xs || hasSum n xs
descProperDivisors
:: Integral a
=> a -> [a]
descProperDivisors n =
let root = (floor . sqrt) (fromIntegral n :: Double)
lows = filter ((0 ==) . rem n) [root,root - 1 .. 1]
factors
| n == root ^ 2 = tail lows
| otherwise = lows
in tail $ reverse (quot n <$> lows) ++ factors
main :: IO ()
main =
(putStrLn . unlines) $
zipWith (\i x -> show i ++ (" -> " ++ show x)) [1 ..] (take 25 weirds)
- Output:
1 -> 70 2 -> 836 3 -> 4030 4 -> 5830 5 -> 7192 6 -> 7912 7 -> 9272 8 -> 10430 9 -> 10570 10 -> 10792 11 -> 10990 12 -> 11410 13 -> 11690 14 -> 12110 15 -> 12530 16 -> 12670 17 -> 13370 18 -> 13510 19 -> 13790 20 -> 13930 21 -> 14770 22 -> 15610 23 -> 15890 24 -> 16030 25 -> 16310
JavaScript[edit]
ES6[edit]
(() => {
'use strict';
// main :: IO ()
const main = () =>
take(25, weirds());
// weirds :: Gen [Int]
function* weirds() {
let
x = 1,
i = 1;
while (true) {
x = until(isWeird, succ, x)
console.log(i.toString() + ' -> ' + x)
yield x;
x = 1 + x;
i = 1 + i;
}
}
// isWeird :: Int -> Bool
const isWeird = n => {
const
ds = descProperDivisors(n),
d = sum(ds) - n;
return 0 < d && !hasSum(d, ds)
};
// hasSum :: Int -> [Int] -> Bool
const hasSum = (n, xs) => {
const go = (n, xs) =>
0 < xs.length ? (() => {
const
h = xs[0],
t = xs.slice(1);
return n < h ? (
go(n, t)
) : (
n == h || hasSum(n - h, t) || hasSum(n, t)
);
})() : false;
return go(n, xs);
};
// descProperDivisors :: Int -> [Int]
const descProperDivisors = n => {
const
rRoot = Math.sqrt(n),
intRoot = Math.floor(rRoot),
blnPerfect = rRoot === intRoot,
lows = enumFromThenTo(intRoot, intRoot - 1, 1)
.filter(x => (n % x) === 0);
return (
reverse(lows)
.slice(1)
.map(x => n / x)
).concat((blnPerfect ? tail : id)(lows))
};
// GENERIC FUNCTIONS ----------------------------
// enumFromThenTo :: Int -> Int -> Int -> [Int]
const enumFromThenTo = (x1, x2, y) => {
const d = x2 - x1;
return Array.from({
length: Math.floor(y - x2) / d + 2
}, (_, i) => x1 + (d * i));
};
// id :: a -> a
const id = x => x;
// reverse :: [a] -> [a]
const reverse = xs =>
'string' !== typeof xs ? (
xs.slice(0).reverse()
) : xs.split('').reverse().join('');
// succ :: Enum a => a -> a
const succ = x => 1 + x;
// sum :: [Num] -> Num
const sum = xs => xs.reduce((a, x) => a + x, 0);
// tail :: [a] -> [a]
const tail = xs => 0 < xs.length ? xs.slice(1) : [];
// take :: Int -> [a] -> [a]
// take :: Int -> String -> String
const take = (n, xs) =>
'GeneratorFunction' !== xs.constructor.constructor.name ? (
xs.slice(0, n)
) : [].concat.apply([], Array.from({
length: n
}, () => {
const x = xs.next();
return x.done ? [] : [x.value];
}));
// until :: (a -> Bool) -> (a -> a) -> a -> a
const until = (p, f, x) => {
let v = x;
while (!p(v)) v = f(v);
return v;
};
// MAIN ---
return main();
})();
- Output:
1 -> 70 2 -> 836 3 -> 4030 4 -> 5830 5 -> 7192 6 -> 7912 7 -> 9272 8 -> 10430 9 -> 10570 10 -> 10792 11 -> 10990 12 -> 11410 13 -> 11690 14 -> 12110 15 -> 12530 16 -> 12670 17 -> 13370 18 -> 13510 19 -> 13790 20 -> 13930 21 -> 14770 22 -> 15610 23 -> 15890 24 -> 16030 25 -> 16310
Julia[edit]
using Primes
function nosuchsum(revsorted, num)
if sum(revsorted) < num
return true
end
for (i, n) in enumerate(revsorted)
if n > num
continue
elseif n == num
return false
elseif !nosuchsum(revsorted[i+1:end], num - n)
return false
end
end
true
end
function isweird(n)
if n < 70 || isodd(n)
return false
else
f = [one(n)]
for (p, x) in factor(n)
f = reduce(vcat, [f*p^i for i in 1:x], init=f)
end
pop!(f)
return sum(f) > n && nosuchsum(sort(f, rev=true), n)
end
end
function testweird(N)
println("The first $N weird numbers are: ")
count, n = 0, 69
while count < N
if isweird(n)
count += 1
print("$n ")
end
n += 1
end
end
testweird(25)
- Output:
The first 25 weird numbers are: 70 836 4030 5830 7192 7912 9272 10430 10570 10792 10990 11410 11690 12110 12530 12670 13370 13510 13790 13930 14770 15610 15890 16030 16310
Kotlin[edit]
// Version 1.3.21
fun divisors(n: Int): List<Int> {
val divs = mutableListOf(1)
val divs2 = mutableListOf<Int>()
var i = 2
while (i * i <= n) {
if (n % i == 0) {
val j = n / i
divs.add(i)
if (i != j) divs2.add(j)
}
i++
}
divs2.addAll(divs.asReversed())
return divs2
}
fun abundant(n: Int, divs: List<Int>) = divs.sum() > n
fun semiperfect(n: Int, divs: List<Int>): Boolean {
if (divs.size > 0) {
val h = divs[0]
val t = divs.subList(1, divs.size)
if (n < h) {
return semiperfect(n, t)
} else {
return n == h || semiperfect(n-h, t) || semiperfect(n, t)
}
} else {
return false
}
}
fun sieve(limit: Int): BooleanArray {
// false denotes abundant and not semi-perfect.
// Only interested in even numbers >= 2
val w = BooleanArray(limit)
for (i in 2 until limit step 2) {
if (w[i]) continue
val divs = divisors(i)
if (!abundant(i, divs)) {
w[i] = true
} else if (semiperfect(i, divs)) {
for (j in i until limit step i) w[j] = true
}
}
return w
}
fun main() {
val w = sieve(17000)
var count = 0
val max = 25
println("The first 25 weird numbers are:")
var n = 2
while (count < max) {
if (!w[n]) {
print("$n ")
count++
}
n += 2
}
println()
}
- Output:
The first 25 weird numbers are: 70 836 4030 5830 7192 7912 9272 10430 10570 10792 10990 11410 11690 12110 12530 12670 13370 13510 13790 13930 14770 15610 15890 16030 16310
Perl[edit]
use strict;
use feature 'say';
use List::Util 'sum';
use POSIX 'floor';
use Algorithm::Combinatorics 'subsets';
use ntheory <is_prime divisors>;
sub abundant {
my($x) = @_;
my $s = sum( my @l = is_prime($x) ? 1 : grep { $x != $_ } divisors($x) );
$s > $x ? ($s, sort { $b <=> $a } @l) : ();
}
my(@weird,$n);
while () {
$n++;
my ($sum, @div) = abundant($n);
next unless $sum; # Weird number must be abundant, skip it if it isn't.
next if $sum / $n > 1.1; # There aren't any weird numbers with a sum:number ratio greater than 1.08 or so.
if ($n >= 10430 and (! int $n%70) and is_prime(int $n/70)) {
# It's weird. All numbers of the form 70 * (a prime 149 or larger) are weird
} else {
my $next;
my $l = shift @div;
my $iter = subsets(\@div);
while (my $s = $iter->next) {
++$next and last if sum(@$s) == $n - $l;
}
next if $next;
}
push @weird, $n;
last if @weird == 25;
}
say "The first 25 weird numbers:\n" . join ' ', @weird;
- Output:
The first 25 weird numbers: 70 836 4030 5830 7192 7912 9272 10430 10570 10792 10990 11410 11690 12110 12530 12670 13370 13510 13790 13930 14770 15610 15890 16030 16310
Simpler and faster solution:
use 5.010;
use strict;
use ntheory qw(vecsum divisors divisor_sum);
sub is_pseudoperfect {
my ($n, $d, $s, $m) = @_;
$d //= do { my @d = divisors($n); pop(@d); \@d };
$s //= vecsum(@$d);
$m //= $#$d;
return 0 if $m < 0;
while ($d->[$m] > $n) {
$s -= $d->[$m--];
}
return 1 if ($n == $s or $d->[$m] == $n);
is_pseudoperfect($n-$d->[$m], $d, $s-$d->[$m], $m - 1) ||
is_pseudoperfect($n, $d, $s-$d->[$m], $m - 1);
}
sub is_weird {
my ($n) = @_;
divisor_sum($n) > 2*$n and not is_pseudoperfect($n);
}
my @weird;
for (my $k = 1 ; @weird < 25 ; ++$k) {
push(@weird, $k) if is_weird($k);
}
say "The first 25 weird numbers:\n@weird";
- Output:
The first 25 weird numbers: 70 836 4030 5830 7192 7912 9272 10430 10570 10792 10990 11410 11690 12110 12530 12670 13370 13510 13790 13930 14770 15610 15890 16030 16310
Perl 6[edit]
sub abundant (\x) {
my @l = x.is-prime ?? 1 !! flat
1, (2 .. x.sqrt.floor).map: -> \d {
my \y = x div d;
next if y * d !== x;
d !== y ?? (d, y) !! d
};
(my $s = @l.sum) > x ?? ($s, |@l.sort(-*)) !! ();
}
my @weird = (2, 4, {|($_ + 4, $_ + 6)} ... *).map: -> $n {
my ($sum, @div) = $n.&abundant;
next unless $sum; # Weird number must be abundant, skip it if it isn't.
next if $sum / $n > 1.1; # There aren't any weird numbers with a sum:number ratio greater than 1.08 or so.
if $n >= 10430 and ($n %% 70) and ($n div 70).is-prime {
# It's weird. All numbers of the form 70 * (a prime 149 or larger) are weird
} else {
my $next;
my $l = @div.shift;
++$next and last if $_.sum == $n - $l for @div.combinations;
next if $next;
}
$n
}
put "The first 25 weird numbers:\n", @weird[^25];
- Output:
The first 25 weird numbers: 70 836 4030 5830 7192 7912 9272 10430 10570 10792 10990 11410 11690 12110 12530 12670 13370 13510 13790 13930 14770 15610 15890 16030 16310
Phix[edit]
Sufficiently fast that I un-optimised it a bit to make it easier to follow.
function abundant(integer n, sequence divs)
return sum(divs) > n
end function
function semiperfect(integer n, sequence divs)
if length(divs)=0 then return false end if
integer h = divs[1]; divs = divs[2..$]
return n=h
or (n>h and semiperfect(n-h, divs))
or semiperfect(n, divs)
end function
function sieve(integer limit)
-- true denotes abundant and not semi-perfect.
-- only interested in even numbers >= 2
sequence wierd := repeat(true,limit)
for j=6 to limit by 6 do
-- eliminate multiples of 3
wierd[j] = false
end for
for i=2 to limit by 2 do
if wierd[i] then
sequence divs := factors(i,-1)
if not abundant(i,divs) then
wierd[i] = false
elsif semiperfect(i,divs) then
for j=i to limit by i do wierd[j] = false end for
end if
end if
end for
return wierd
end function
--constant MAX = 25, sieve_limit = 16313
constant MAX = 50, sieve_limit = 26533
sequence wierd := sieve(sieve_limit), res = {}
for i=2 to sieve_limit by 2 do
if wierd[i] then
res &= i
if length(res)=MAX then exit end if
end if
end for
printf(1,"The first %d weird numbers are: %v\n",{MAX,res})
- Output:
The first 50 weird numbers are: {70,836,4030,5830,7192,7912,9272,10430,10570,10792,10990,11410,11690,12110,12530,12670,13370,13510,13790,13930,14770,15610,15890,16030,16310,
16730,16870,17272,17570,17990,18410,18830,18970,19390,19670,19810,20510,21490,21770,21910,22190,23170,23590,24290,24430,24710,25130,25690,26110,26530}
Python[edit]
Functional[edit]
The first 50 seem to take c. 300 ms
'''Weird numbers'''
from itertools import islice, repeat
from functools import reduce
from math import sqrt
from time import time
# weirds :: Gen [Int]
def weirds():
'''Generator for weird numbers.
(Abundant, but not semi-perfect)'''
x = 1
while True:
x = until(isWeird)(succ)(x)
yield x
x = 1 + x
# isWeird :: Int -> Bool
def isWeird(n):
'''Predicate :: abundant and not semi-perfect ?'''
ds = descPropDivs(n)
d = sum(ds) - n
return 0 < d and not hasSum(d, ds)
# hasSum :: Int -> [Int] -> Bool
def hasSum(n, xs):
'''Does any subset of xs sum to n ?
(Assuming xs to be sorted in descending
order of magnitude)'''
def go(n, xs):
if xs:
h, t = xs[0], xs[1:]
if n < h: # Head too big. Forget it. Tail ?
return go(n, t)
else:
# The head IS the target ?
# Or the tail contains a sum for the
# DIFFERENCE between the head and the target ?
# Or the tail contains some OTHER sum for the target ?
return n == h or go(n - h, t) or go(n, t)
else:
return False
return go(n, xs)
# descPropDivs :: Int -> [Int]
def descPropDivs(n):
'''Descending positive divisors of n,
excluding n itself.'''
root = sqrt(n)
intRoot = int(root)
blnSqr = root == intRoot
lows = [x for x in range(1, 1 + intRoot) if 0 == n % x]
return [
n // x for x in (
lows[1:-1] if blnSqr else lows[1:]
)
] + list(reversed(lows))
# TEST ----------------------------------------------------
# main :: IO ()
def main():
'''Test'''
start = time()
n = 50
xs = take(n)(weirds())
print(
(tabulated('First ' + str(n) + ' weird numbers:\n')(
lambda i: str(1 + i)
)(str)(5)(
index(xs)
)(range(0, n)))
)
print(
'\nApprox computation time: ' +
str(int(1000 * (time() - start))) + ' ms'
)
# GENERIC -------------------------------------------------
# chunksOf :: Int -> [a] -> [[a]]
def chunksOf(n):
'''A series of lists of length n,
subdividing the contents of xs.
Where the length of xs is not evenly divible,
the final list will be shorter than n.'''
return lambda xs: reduce(
lambda a, i: a + [xs[i:n + i]],
range(0, len(xs), n), []
) if 0 < n else []
# compose (<<<) :: (b -> c) -> (a -> b) -> a -> c
def compose(g):
'''Right to left function composition.'''
return lambda f: lambda x: g(f(x))
# index (!!) :: [a] -> Int -> a
def index(xs):
'''Item at given (zero-based) index.'''
return lambda n: None if 0 > n else (
xs[n] if (
hasattr(xs, "__getitem__")
) else next(islice(xs, n, None))
)
# paddedMatrix :: a -> [[a]] -> [[a]]
def paddedMatrix(v):
''''A list of rows padded to equal length
(where needed) with instances of the value v.'''
def go(rows):
return paddedRows(
len(max(rows, key=len))
)(v)(rows)
return lambda rows: go(rows) if rows else []
# paddedRows :: Int -> a -> [[a]] -[[a]]
def paddedRows(n):
'''A list of rows padded (but never truncated)
to length n with copies of value v.'''
def go(v, xs):
def pad(x):
d = n - len(x)
return (x + list(repeat(v, d))) if 0 < d else x
return list(map(pad, xs))
return lambda v: lambda xs: go(v, xs) if xs else []
# showColumns :: Int -> [String] -> String
def showColumns(n):
'''A column-wrapped string
derived from a list of rows.'''
def go(xs):
def fit(col):
w = len(max(col, key=len))
def pad(x):
return x.ljust(4 + w, ' ')
return ''.join(map(pad, col))
q, r = divmod(len(xs), n)
return unlines(map(
fit,
transpose(paddedMatrix('')(
chunksOf(q + int(bool(r)))(
xs
)
))
))
return lambda xs: go(xs)
# succ :: Enum a => a -> a
def succ(x):
'''The successor of a value. For numeric types, (1 +).'''
return 1 + x if isinstance(x, int) else (
chr(1 + ord(x))
)
# tabulated :: String -> (a -> String) ->
# (b -> String) ->
# Int ->
# (a -> b) -> [a] -> String
def tabulated(s):
'''Heading -> x display function -> fx display function ->
number of columns -> f -> value list -> tabular string.'''
def go(xShow, fxShow, intCols, f, xs):
w = max(map(compose(len)(xShow), xs))
return s + '\n' + showColumns(intCols)([
xShow(x).rjust(w, ' ') + ' -> ' + fxShow(f(x)) for x in xs
])
return lambda xShow: lambda fxShow: lambda nCols: (
lambda f: lambda xs: go(
xShow, fxShow, nCols, f, xs
)
)
# take :: Int -> [a] -> [a]
# take :: Int -> String -> String
def take(n):
'''The prefix of xs of length n,
or xs itself if n > length xs.'''
return lambda xs: (
xs[0:n]
if isinstance(xs, list)
else list(islice(xs, n))
)
# transpose :: Matrix a -> Matrix a
def transpose(m):
'''The rows and columns of the argument transposed.
(The matrix containers and rows can be lists or tuples).'''
if m:
inner = type(m[0])
z = zip(*m)
return (type(m))(
map(inner, z) if tuple != inner else z
)
else:
return m
# unlines :: [String] -> String
def unlines(xs):
'''A single string derived by the intercalation
of a list of strings with the newline character.'''
return '\n'.join(xs)
# until :: (a -> Bool) -> (a -> a) -> a -> a
def until(p):
'''The result of repeatedly applying f until p holds.
The initial seed value is x.'''
def go(f, x):
v = x
while not p(v):
v = f(v)
return v
return lambda f: lambda x: go(f, x)
# MAIN ----------------------------------------------------
if __name__ == '__main__':
main()
- Output:
First 50 weird numbers: 1 -> 70 11 -> 10990 21 -> 14770 31 -> 18410 41 -> 22190 2 -> 836 12 -> 11410 22 -> 15610 32 -> 18830 42 -> 23170 3 -> 4030 13 -> 11690 23 -> 15890 33 -> 18970 43 -> 23590 4 -> 5830 14 -> 12110 24 -> 16030 34 -> 19390 44 -> 24290 5 -> 7192 15 -> 12530 25 -> 16310 35 -> 19670 45 -> 24430 6 -> 7912 16 -> 12670 26 -> 16730 36 -> 19810 46 -> 24710 7 -> 9272 17 -> 13370 27 -> 16870 37 -> 20510 47 -> 25130 8 -> 10430 18 -> 13510 28 -> 17272 38 -> 21490 48 -> 25690 9 -> 10570 19 -> 13790 29 -> 17570 39 -> 21770 49 -> 26110 10 -> 10792 20 -> 13930 30 -> 17990 40 -> 21910 50 -> 26530 Approx computation time: 278 ms
REXX[edit]
This REXX program could be optimized by finding and using primitive weird numbers and multiplying them by
prime numbers ≥ 149 to find higher weird numbers, but it would've added complexity to the REXX program.
/*REXX program finds and displays N weird numbers in a vertical format (with index).*/
parse arg n . /*obtain optional arguments from the CL*/
if n=='' | n=="," then n= 25 /*Not specified? Then use the default.*/
#= 0 /*the count of weird numbers (so far).*/
do j=2 by 2 until #==n /*examine even integers 'til have 'nuff*/
if \weird(j) then iterate /*Not a weird number? Then skip it. */
#= # + 1 /*bump the count of weird numbers. */
say right(th(#), 30) ' weird number is:' right(commas(j), 9) /*display weird #.*/
end /*j*/
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
commas: parse arg _; do ic=length(_)-3 to 1 by -3; _=insert(',', _, ic); end; return _
th: parse arg th;return th||word('th st nd rd',1+(th//10)*(th//100%10\==1)*(th//10<4))
/*──────────────────────────────────────────────────────────────────────────────────────*/
DaS: procedure; parse arg x 1 z 1,b; a= 1 /*get X,Z,B (the 1st arg); init A list.*/
r= 0; q= 1 /* [↓] ══integer square root══ ___ */
do while q<=z; q=q*4; end /*R: an integer which will be √ X */
do while q>1; q=q%4; _= z-r-q; r=r%2; if _>=0 then do; z=_; r=r+q; end
end /*while q>1*/ /* [↑] compute the integer sqrt of X.*/
sig = a /*initialize the sigma so far. ___ */
do j=2 to r - (r*r==x) /*divide by some integers up to √ X */
if x//j==0 then do; a=a j; b=x%j b /*if ÷, add both divisors to α & ß. */
sig= sig+j+ x%j /*bump the sigma (the sum of divisors).*/
end
end /*j*/ /* [↑] % is the REXX integer division*/
/* [↓] adjust for a square. ___*/
if j*j==x then return sig+j a j b /*Was X a square? If so, add √ X */
return sig a b /*return the divisors (both lists). */
/*──────────────────────────────────────────────────────────────────────────────────────*/
weird: procedure; parse arg x . /*obtain a # to be tested for weirdness*/
if x<70 | x//3==0 then return 0 /*test if X is too low or multiple of 3*/
parse value DaS(x) with sigma divs /*obtain sigma and the proper divisors.*/
if sigma<=x then return 0 /*X isn't abundant (sigma too small).*/
#= words(divs) /*count the number of divisors for X. */
if #<3 then return 0 /*Not enough divisors? " " */
if #>15 then return 0 /*number of divs > 15? It's not weird.*/
a.= /*initialize the A. stemmed array.*/
do i=1 for #; _=word(divs, i) /*obtain one of the divisors of X. */
@.i= _; a._=. /*assign proper divs──►@ array; also id*/
end /*i*/
df= sigma - x /*calculate difference between Σ and X.*/
if a.df==. then return 0 /*Any divisor is equal to DF? Not weird*/
c=0; u= 2**# /*zero combo counter; calc. power of 2.*/
do p=1 for u-2; c= c + 1 /*convert P──►binary with leading zeros*/
yy.c= strip( x2b( d2x(p) ), 'L', 0) /*store this particular combination. */
end /*p*/
/* [↓] decreasing partitions is faster*/
do part=c by -1 for c; s= 0 /*test of a partition add to the arg X.*/
_= yy.part; L=length(_) /*obtain one method of partitioning. */
do cp=L by -1 for L /*obtain a sum of a partition. */
if substr(_,cp,1) then do; s= s + @.cp /*1 bit? Then add ──►S*/
if s==x then return 0 /*Sum equal? Not weird*/
if s==df then return 0 /*Sum = DF? " " */
if s>x then iterate /*Sum too big? Try next*/
end
end /*cp*/
end /*part*/
return 1 /*no sum equal to X, so X is weird.*/
- output when using the input of: 50
1st weird number is: 70
2nd weird number is: 836
3rd weird number is: 4,030
4th weird number is: 5,830
5th weird number is: 7,192
6th weird number is: 7,912
7th weird number is: 9,272
8th weird number is: 10,430
9th weird number is: 10,570
10th weird number is: 10,792
11th weird number is: 10,990
12th weird number is: 11,410
13th weird number is: 11,690
14th weird number is: 12,110
15th weird number is: 12,530
16th weird number is: 12,670
17th weird number is: 13,370
18th weird number is: 13,510
19th weird number is: 13,790
20th weird number is: 13,930
21st weird number is: 14,770
22nd weird number is: 15,610
23rd weird number is: 15,890
24th weird number is: 16,030
25th weird number is: 16,310
26th weird number is: 16,730
27th weird number is: 16,870
28th weird number is: 17,272
29th weird number is: 17,570
30th weird number is: 17,990
31st weird number is: 18,410
32nd weird number is: 18,830
33rd weird number is: 18,970
34th weird number is: 19,390
35th weird number is: 19,670
36th weird number is: 19,810
37th weird number is: 20,510
38th weird number is: 21,490
39th weird number is: 21,770
40th weird number is: 21,910
41st weird number is: 22,190
42nd weird number is: 23,170
43rd weird number is: 23,590
44th weird number is: 24,290
45th weird number is: 24,430
46th weird number is: 24,710
47th weird number is: 25,130
48th weird number is: 25,690
49th weird number is: 26,110
50th weird number is: 26,530
Sidef[edit]
func is_pseudoperfect(n, d = n.divisors.slice(0, -2), s = d.sum, m = d.end) {
return false if (m < 0)
while (d[m] > n) {
s -= d[m--]
}
return true if (n == s)
return true if (d[m] == n)
__FUNC__(n-d[m], d, s-d[m], m-1) || __FUNC__(n, d, s-d[m], m-1)
}
func is_weird(n) {
(n.sigma > 2*n) && !is_pseudoperfect(n)
}
var w = (1..Inf -> lazy.grep(is_weird).first(25))
say "The first 25 weird numbers:\n#{w.join(' ')}"
- Output:
The first 25 weird numbers: 70 836 4030 5830 7192 7912 9272 10430 10570 10792 10990 11410 11690 12110 12530 12670 13370 13510 13790 13930 14770 15610 15890 16030 16310
Visual Basic .NET[edit]
Performance is now on par with the python version, (but not quite up the Go version's performance), I applied what I could after reading the comments made by Hout on the discussion page.
This program is similar to the structure of the Go example. I found a couple of tweaks here and there to help with performance. For example, the divisors list is built on a single array instead of joining two, and it calculates the sum while creating the divisors list. The divisors list is headed by the difference between "n" and the sum of the divisors. The semiperfect() function checks for equality first (rather than chopping the head from the tail list first) to save a little more time. And of course, the parallel execution.
A new feature is that one can calculate weird numbers up to any reasonable number, just enter a command line parameter of more than zero. Another new feature is calculating weird numbers continuously until a key is pressed (like the spigot algorithm from the Pi task) - to do so, enter a command line parameter of less than 1.
This has no sieve cache, as one must "know" beforehand what number to cache up to, (for best results). Since there is no cache (runs slower), I added the parallel execution to make it run faster.
I haven't let it run long enough to see how high it can get before crashing, I suspect it should happen once the weird number being tested is around Int32.MaxValue (2,147,483,647). But long before that it will slow down quite a bit. It takes around 17 minutes to get to the 10,732nd weird number, which is the first over 7 million (7,000,210).
Module Module1
Dim resu As New List(Of Integer)
Function TestAbundant(n As Integer, ByRef divs As List(Of Integer)) As Boolean
divs = New List(Of Integer)
Dim sum As Integer = -n : For i As Integer = Math.Sqrt(n) To 1 Step -1
If n Mod i = 0 Then divs.Add(i) : Dim j As Integer = n / i : divs.Insert(0, j) : sum += i + j
Next : divs(0) = sum - divs(0) : Return divs(0) > 0
End Function
Function subList(src As List(Of Integer), Optional first As Integer = Integer.MinValue) As List(Of Integer)
subList = src.ToList : subList.RemoveAt(1)
End Function
Function semiperfect(divs As List(Of Integer)) As Boolean
If divs.Count < 2 Then Return False
Select Case divs.First.CompareTo(divs(1))
Case 0 : Return True
Case -1 : Return semiperfect(subList(divs))
Case 1 : Dim t As List(Of Integer) = subList(divs) : t(0) -= divs(1)
If semiperfect(t) Then Return True Else t(0) = divs.First : Return semiperfect(t)
End Select : Return False ' execution can't get here, just for compiler warning
End Function
Function Since(et As TimeSpan) As String ' big ugly routine to prettify the elasped time
If et > New TimeSpan(2000000) Then
Dim s As String = " " & et.ToString(), p As Integer = s.IndexOf(":"), q As Integer = s.IndexOf(".")
If q < p Then s = s.Insert(q, "Days") : s = s.Replace("Days.", "Days, ")
p = s.IndexOf(":") : s = s.Insert(p, "h") : s = s.Replace("h:", "h ")
p = s.IndexOf(":") : s = s.Insert(p, "m") : s = s.Replace("m:", "m ")
s = s.Replace(" 0", " ").Replace(" 0h", " ").Replace(" 0m", " ") & "s"
Return s.TrimStart()
Else
If et > New TimeSpan(1500) Then
Return et.TotalMilliseconds.ToString() & "ms"
Else
If et > New TimeSpan(15) Then
Return (et.TotalMilliseconds * 1000.0).ToString() & "µs"
Else
Return (et.TotalMilliseconds * 1000000.0).ToString() & "ns"
End If
End If
End If
End Function
Sub Main(args As String())
Dim sw As New Stopwatch, st As Integer = 2, stp As Integer = 1020, count As Integer = 0
Dim max As Integer = 25, halted As Boolean = False
If args.Length > 0 Then _
Dim t As Integer = Integer.MaxValue : If Integer.TryParse(args(0), t) Then max = If(t > 0, t, Integer.MaxValue)
If max = Integer.MaxValue Then
Console.WriteLine("Calculating weird numbers, press a key to halt.")
stp *= 10
Else
Console.WriteLine("The first {0} weird numbers:", max)
End If
If max < 25 Then stp = 140
sw.Start()
Do : Parallel.ForEach(Enumerable.Range(st, stp),
Sub(n)
Dim divs As List(Of Integer) = Nothing
If TestAbundant(n, divs) AndAlso Not semiperfect(divs) Then
SyncLock resu : resu.Add(n) : End SyncLock
End If
End Sub)
If resu.Count > 0 Then
resu.Sort()
If count + resu.Count > max Then
resu = resu.Take(max - count).ToList
End If
Console.Write(String.Join(" ", resu) & " ")
count += resu.Count : resu.Clear()
End If
If Console.KeyAvailable Then Console.ReadKey() : halted = True : Exit Do
st += stp
Loop Until count >= max
sw.Stop()
If max < Integer.MaxValue Then
Console.WriteLine(vbLf & "Computation time was {0}.", Since(sw.Elapsed))
If halted Then Console.WriteLine("Halted at number {0}.", count)
Else
Console.WriteLine(vbLf & "Computation time was {0} for the first {1} weird numbers.", Since(sw.Elapsed), count)
End If
End Sub
End Module
- Output:
Without any command line parameters:
The first 25 weird numbers: 70 836 4030 5830 7192 7912 9272 10430 10570 10792 10990 11410 11690 12110 12530 12670 13370 13510 13790 13930 14770 15610 15890 16030 16310 Computation time was 37.4931ms.
With command line arguments = 50
The first 50 weird numbers: 70 836 4030 5830 7192 7912 9272 10430 10570 10792 10990 11410 11690 12110 12530 12670 13370 13510 13790 13930 14770 15610 15890 16030 16310 16730 16870 17272 17570 17990 18410 18830 18970 19390 19670 19810 20510 21490 21770 21910 22190 23170 23590 24290 24430 24710 25130 25690 26110 26530 Computation time was 47.6589ms.
With command line arguments = 0
Calculating weird numbers, press a key to halt. 70 836 4030 5830 7192 7912 9272 10430 10570 10792 10990 11410 11690 12110 12530 12670 13370 13510 13790 13930 14770 15610 15890 16030 16310 16730 16870 17272 17570 17990 18410 18830 18970 19390 19670 19810 20510 21490 21770 21910 22190 23170 23590 24290 24430 24710 25130 25690 26110 26530 26810 27230 27790 28070 28630 29330 29470 30170 30310 30730 31010 31430 31990 32270 32410 32690 33530 34090 34370 34930 35210 35630 36470 36610 37870 38290 38990 39410 39830 39970 40390 41090 41510 41930 42070 42490 42910 43190 43330 44170 44870 45010 45290 45356 45710 46130 46270 47110 47390 47810 48370 49070 49630 50330 50890 51310 51730 52010 52570 52990 53270 53830 54110 55090 55790 56630 56770 57470 57610 57890 58030 58730 59710 59990 60130 60410 61390 61670 61810 62090 63490 63770 64330 65030 65590 65870 66290 66710 67690 67970 68390 68810 69370 69790 70630 70910 71330 71470 72170 72310 72730 73430 73570 73616 74270 74410 74830 76090 76370 76510 76790 77210 77630 78190 78610 79030 80570 80710 81410 81970 82670 83090 83312 83510 84070 84910 85190 85610 86030 86170 86590 87430 88130 89390 89530 89810 90230 90370 90790 91070 91210 91388 91490 92330 92470 92890 95270 95690 96110 96670 97930 98630 99610 99890 100030 100310 100730 101290 101570 101710 102130 102970 103670 103810 104090 104230 104510 104930 105770 106610 107170 108010 108430 108710 109130 109690 109970 110530 110810 111790 112070 112490 112630 112910 113072 113330 113470 113890 114590 115990 116410 116690 116830 118510 118790 118930 119630 120470 120610 121310 121870 122290 122710 123130 124390 124810 125090 125230 126070 126770 127610 128170 129290 130270 130690 130970 131110 131390 131530 132230 133070 133490 133910 135170 135310 136430 136570 138110 138530 139090 139510 139790 139930 140210 140770 Computation time was 153.3649ms for the first 285 weird numbers.
Tail-end of a longer session:
6981310 6983108 6983270 6983690 6985090 6985510 6986630 6987190 6987610 6988030 6988310 6988730 6990130 6990970 6991390 6991468 6991670 6992930 6993070 6993490 6994610 6995030 6996484 6997270 6997970 6998110 6999230 6999370 7000210 7001330 7003010 7003172 7003430 7003990 7004830 7007210 7007630 7008890 7009030 Computation time was 17m 9.0062776s for the first 10742 weird numbers.
