Weird numbers

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

AppleScript

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).

<lang applescript>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</lang>

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}

Go

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. <lang go>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()

}</lang>

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

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

Translation of: Python

<lang haskell>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]
 in tail $
    reverse (quot n <$> lows) ++
    (if n == root * root
       then tail
       else id)
      lows

main :: IO () main =

 (putStrLn . unlines) $
 zipWith (\i x -> show i ++ (" -> " ++ show x)) [1 ..] (take 25 weirds)</lang>

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

ES6

Translation of: Python

Translation of: Haskell

<lang JavaScript>(() => {

   '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();

})();</lang>

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

<lang Julia>using Primes

function canmakesum(revsorted, num)

   ret = Int[]
   for (i, n) in enumerate(revsorted[1:end])
       if n > num
           continue
       elseif n == num
           return [n]
       else
           arr = canmakesum(revsorted[i+1:end], num - n)
           if arr != ret
               return vcat([n], arr)
           end
       end
   end
   ret

end

function isweird(n)

   if n < 70
       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 && canmakesum(sort(f, rev=true), n) == Int[]
   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)

</lang>

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

Translation of: Go

<lang scala>// 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()

}</lang>

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

Translation of: Perl 6

Library: ntheory

<lang perl>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;</lang>

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:

Translation of: Sidef

Library: ntheory

<lang perl>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";</lang>

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

<lang perl6>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];</lang>

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

Python

Functional

The first 50 seem to take c. 300 ms

Works with: Python version 3

<lang python>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()</lang>

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

Sidef

<lang ruby>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(' ')}"</lang>

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

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). <lang vbnet>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</lang>

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.