Weird numbers
In number theory, a weird number is a natural number that is abundant but not semiperfect (and therefore not perfect either).
You are encouraged to solve this task according to the task description, using any language you may know.
In other words, the sum of the proper divisors of the number (divisors including 1 but not itself) is greater than the number itself (the number is abundant), but no subset of those divisors sums to the number itself (the number is not semiperfect).
For example:
- 12 is not a weird number.
- It is abundant; its proper divisors 1, 2, 3, 4, 6 sum to 16 (which is > 12),
- but it is semiperfect, e.g.: 6 + 4 + 2 == 12.
- 70 is a weird number.
- It is abundant; its proper divisors 1, 2, 5, 7, 10, 14, 35 sum to 74 (which is > 70),
- and there is no subset of proper divisors that sum to 70.
- Task
Find and display, here on this page, the first 25 weird numbers.
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}
C
<lang c>#include "stdio.h"
- include "stdlib.h"
- include "stdbool.h"
- include "string.h"
struct int_a {
int *ptr; size_t size;
};
struct int_a divisors(int n) {
int *divs, *divs2, *out; int i, j, c1 = 0, c2 = 0; struct int_a array;
divs = malloc(n * sizeof(int) / 2); divs2 = malloc(n * sizeof(int) / 2); divs[c1++] = 1;
for (i = 2; i * i <= n; i++) {
if (n % i == 0) {
j = n / i;
divs[c1++] = i;
if (i != j) {
divs2[c2++] = j;
}
}
}
out = malloc((c1 + c2) * sizeof(int));
for (int i = 0; i < c2; i++) {
out[i] = divs2[i];
}
for (int i = 0; i < c1; i++) {
out[c2 + i] = divs[c1 - i - 1];
}
array.ptr = out;
array.size = c1 + c2;
free(divs); free(divs2); return array;
}
bool abundant(int n, struct int_a divs) {
int sum = 0;
int i;
for (i = 0; i < divs.size; i++) {
sum += divs.ptr[i];
}
return sum > n;
}
bool semiperfect(int n, struct int_a divs) {
if (divs.size > 0) {
int h = *divs.ptr;
int *t = divs.ptr + 1;
struct int_a ta;
ta.ptr = t;
ta.size = divs.size - 1;
if (n < h) {
return semiperfect(n, ta);
} else {
return n == h
|| semiperfect(n - h, ta)
|| semiperfect(n, ta);
}
} else {
return false;
}
}
bool *sieve(int limit) {
bool *w = calloc(limit, sizeof(bool)); struct int_a divs; int i, j;
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;
}
}
}
free(divs.ptr); return w;
}
int main() {
bool *w = sieve(17000); int count = 0; int max = 25; int n;
printf("The first 25 weird numbers:\n");
for (n = 2; count < max; n += 2) {
if (!w[n]) {
printf("%d ", n);
count++;
}
}
printf("\n");
free(w); return 0;
}</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
C#
<lang csharp>using System; using System.Collections.Generic; using System.Linq; using System.Text; using System.Threading.Tasks;
namespace WeirdNumbers {
class Program {
static List<int> Divisors(int n) {
List<int> divs = new List<int> { 1 };
List<int> divs2 = new List<int>();
for (int i = 2; i * i <= n; i++) {
if (n % i == 0) {
int j = n / i;
divs.Add(i);
if (i != j) {
divs2.Add(j);
}
}
}
divs.Reverse();
divs2.AddRange(divs);
return divs2;
}
static bool Abundant(int n, List<int> divs) {
return divs.Sum() > n;
}
static bool Semiperfect(int n, List<int> divs) {
if (divs.Count > 0) {
var h = divs[0];
var t = divs.Skip(1).ToList();
if (n < h) {
return Semiperfect(n, t);
} else {
return n == h
|| Semiperfect(n - h, t)
|| Semiperfect(n, t);
}
} else {
return false;
}
}
static List<bool> Sieve(int limit) {
// false denotes abundant and not semi-perfect.
// Only interested in even numbers >= 2
bool[] w = new bool[limit];
for (int i = 2; i < limit; i += 2) {
if (w[i]) continue;
var 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.ToList();
}
static void Main() {
var w = Sieve(17_000);
int count = 0;
int max = 25;
Console.WriteLine("The first 25 weird numbers:");
for (int n = 2; count < max; n += 2) {
if (!w[n]) {
Console.Write("{0} ", n);
count++;
}
}
Console.WriteLine();
}
}
}</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
C++
<lang cpp>#include <algorithm>
- include <iostream>
- include <numeric>
- include <vector>
std::vector<int> divisors(int n) {
std::vector<int> divs = { 1 };
std::vector<int> divs2;
for (int i = 2; i * i <= n; i++) {
if (n % i == 0) {
int j = n / i;
divs.push_back(i);
if (i != j) {
divs2.push_back(j);
}
}
}
std::copy(divs.cbegin(), divs.cend(), std::back_inserter(divs2)); return divs2;
}
bool abundant(int n, const std::vector<int> &divs) {
return std::accumulate(divs.cbegin(), divs.cend(), 0) > n;
}
template<typename IT> bool semiperfect(int n, const IT &it, const IT &end) {
if (it != end) {
auto h = *it;
auto t = std::next(it);
if (n < h) {
return semiperfect(n, t, end);
} else {
return n == h
|| semiperfect(n - h, t, end)
|| semiperfect(n, t, end);
}
} else {
return false;
}
}
template<typename C> bool semiperfect(int n, const C &c) {
return semiperfect(n, std::cbegin(c), std::cend(c));
}
std::vector<bool> sieve(int limit) {
// false denotes abundant and not semi-perfect.
// Only interested in even numbers >= 2
std::vector<bool> w(limit);
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;
}
int main() {
auto w = sieve(17000);
int count = 0;
int max = 25;
std::cout << "The first 25 weird numbers:";
for (int n = 2; count < max; n += 2) {
if (!w[n]) {
std::cout << n << ' ';
count++;
}
}
std::cout << '\n';
return 0;
}</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
Crystal
<lang ruby>def divisors(n : Int32) : Array(Int32)
divs = [1] divs2 = [] of Int32
i = 2
while i * i < n
if n % i == 0
j = n // i
divs << i
divs2 << j if i != j
end
i += 1 end
i = divs.size - 1
# TODO: Use reverse while i >= 0 divs2 << divs[i] i -= 1 end
divs2
end
def abundant(n : Int32, divs : Array(Int32)) : Bool
divs.sum > n
end
def semiperfect(n : Int32, divs : Array(Int32)) : Bool
if divs.size > 0 h = divs[0] t = divs[1..]
return n < h ? semiperfect(n, t) : n == h || semiperfect(n - h, t) || semiperfect(n, t) end
return false
end
def sieve(limit : Int32) : Array(Bool)
# false denotes abundant and not semi-perfect. # Only interested in even numbers >= 2
w = Array(Bool).new(limit, false) # An array filled with 'false'
i = 2
while i < limit
if !w[i]
divs = divisors i
if !abundant(i, divs)
w[i] = true
elsif semiperfect(i, divs)
j = i
while j < limit
w[j] = true
j += i
end
end
end
i += 2 end
w
end
def main
w = sieve 17000 count = 0 max = 25
print "The first 25 weird numbers are: "
n = 2
while count < max
if !w[n]
print "#{n} "
count += 1
end
n += 2 end
puts "\n"
end
require "benchmark" puts Benchmark.measure { main } </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 # Benchmark with --release flag 0.046875 0.000000 0.046875 ( 0.040754)
D
<lang d>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;
}</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
F#
<lang fsharp>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</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
Factor
The has-sum? word is a translation of the Haskell function.
<lang factor>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</lang>
- 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
Version 1
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 less than 10 ms on an Intel Core i7-8565U machine. 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), which was supplied by Enter your username, is almost 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
Version 2 (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
<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]
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)</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
J
This algorithm uses a sieve to eliminate multiples of semiperfect numbers from future testing. <lang> factor=: [: }: [: , [: */&> [: { [: <@(^ i.@>:)/"1 [: |: __&q:
classify=: 3 : 0
weird =: perfect =: deficient =: abundant =: i. 0
a=: (i. -. 0 , deficient =: 1 , i.&.:(p:inv)) y NB. a are potential semi-perfect numbers
for_n. a do.
if. n e. a do.
factors=. factor n
sf =. +/ factors
if. sf < n do.
deficient =: deficient , n
else.
if. n < sf do.
abundant=: abundant , n
else.
perfect =: perfect , n
a =: a -. (2+i.)@<.&.(%&n) y NB. remove multiples of perfect numbers
continue.
end.
NB. compute sums of subsets to detect semiperfection
NB. the following algorithm correctly finds weird numbers less than 20000
NB. remove large terms necessary for the sum to reduce the Catalan tally of sets
factors =. /:~ factors NB. ascending sort
NB. if the sum of the length one outfixes is less n then the factor is required in the semiperfect set.
i_required =. n (1 i.~ (>(1+/\.]))) factors
target =. n - +/ i_required }. factors
t =. i_required {. factors
NB. work in chunks of 2^16 to reduce memory requirement
sp =. target e. ; (,:~2^16) <@([: +/"1 t #~ (_ ,(#t)) {. #:);.3 i. 2 ^ # t
if. sp do.
a =: a -. (2+i.)@<.&.(%&n) y NB. remove multiples of semi perfect numbers
else.
weird =: weird , n
a =: a -. n
end.
end.
end.
end.
a =: a -. deficient
weird
) </lang>
classify 20000 NB. the first 36 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
Java
<lang java> import java.util.ArrayList; import java.util.List;
public class WeirdNumbers {
public static void main(String[] args) {
int n = 2;
// n += 2 : No odd weird numbers < 10^21
for ( int count = 1 ; count <= 25 ; n += 2 ) {
if ( isWeird(n) ) {
System.out.printf("w(%d) = %d%n", count, n);
count++;
}
}
}
private static boolean isWeird(int n) {
List<Integer> properDivisors = getProperDivisors(n);
return isAbundant(properDivisors, n) && ! isSemiPerfect(properDivisors, n);
}
private static boolean isAbundant(List<Integer> divisors, int n) {
int divisorSum = divisors.stream().mapToInt(i -> i.intValue()).sum();
return divisorSum > n;
}
// Use Dynamic Programming
private static boolean isSemiPerfect(List<Integer> divisors, int sum) {
int size = divisors.size();
// The value of subset[i][j] will be true if there is a subset of divisors[0..j-1] with sum equal to i
boolean subset[][] = new boolean[sum+1][size+1];
// If sum is 0, then answer is true
for (int i = 0; i <= size; i++) {
subset[0][i] = true;
}
// If sum is not 0 and set is empty, then answer is false
for (int i = 1; i <= sum; i++) {
subset[i][0] = false;
}
// Fill the subset table in bottom up manner
for ( int i = 1 ; i <= sum ; i++ ) {
for ( int j = 1 ; j <= size ; j++ ) {
subset[i][j] = subset[i][j-1];
int test = divisors.get(j-1);
if ( i >= test ) {
subset[i][j] = subset[i][j] || subset[i - test][j-1];
}
}
}
return subset[sum][size]; }
private static final List<Integer> getProperDivisors(int number) {
List<Integer> divisors = new ArrayList<Integer>();
long sqrt = (long) Math.sqrt(number);
for ( int i = 1 ; i <= sqrt ; i++ ) {
if ( number % i == 0 ) {
divisors.add(i);
int div = number / i;
if ( div != i && div != number ) {
divisors.add(div);
}
}
}
return divisors;
}
} </lang>
- Output:
w(1) = 70 w(2) = 836 w(3) = 4030 w(4) = 5830 w(5) = 7192 w(6) = 7912 w(7) = 9272 w(8) = 10430 w(9) = 10570 w(10) = 10792 w(11) = 10990 w(12) = 11410 w(13) = 11690 w(14) = 12110 w(15) = 12530 w(16) = 12670 w(17) = 13370 w(18) = 13510 w(19) = 13790 w(20) = 13930 w(21) = 14770 w(22) = 15610 w(23) = 15890 w(24) = 16030 w(25) = 16310
JavaScript
ES6
<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 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)
</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
<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
Lua
<lang lua>function make(n, d)
local a = {}
for i=1,n do
table.insert(a, d)
end
return a
end
function reverse(t)
local n = #t
local i = 1
while i < n do
t[i],t[n] = t[n],t[i]
i = i + 1
n = n - 1
end
end
function tail(list)
return { select(2, unpack(list)) }
end
function divisors(n)
local divs = {}
table.insert(divs, 1)
local divs2 = {}
local i = 2
while i * i <= n do
if n % i == 0 then
local j = n / i
table.insert(divs, i)
if i ~= j then
table.insert(divs2, j)
end
end
i = i + 1
end
reverse(divs)
for i,v in pairs(divs) do
table.insert(divs2, v)
end
return divs2
end
function abundant(n, divs)
local sum = 0
for i,v in pairs(divs) do
sum = sum + v
end
return sum > n
end
function semiPerfect(n, divs)
if #divs > 0 then
local h = divs[1]
local t = tail(divs)
if n < h then
return semiPerfect(n, t)
else
return n == h
or semiPerfect(n - h, t)
or semiPerfect(n, t)
end
else
return false
end
end
function sieve(limit)
-- false denotes abundant and not semi-perfect.
-- Only interested in even numbers >= 2
local w = make(limit, false)
local i = 2
while i < limit do
if not w[i] then
local divs = divisors(i)
if not abundant(i, divs) then
w[i] = true
elseif semiPerfect(i, divs) then
local j = i
while j < limit do
w[j] = true
j = j + i
end
end
end
i = i + 1
end
return w
end
function main()
local w = sieve(17000)
local count = 0
local max = 25
print("The first 25 weird numbers:")
local n = 2
while count < max do
if not w[n] then
io.write(n, ' ')
count = count + 1
end
n = n + 2
end
print()
end
main()</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
Nim
<lang Nim>import algorithm, math, strutils
func divisors(n: int): seq[int] =
var smallDivs = @[1]
for i in 2..sqrt(n.toFloat).int:
if n mod i == 0:
let j = n div i
smallDivs.add i
if i != j: result.add j
result.add reversed(smallDivs)
func abundant(n: int; divs: seq[int]): bool {.inline.}=
sum(divs) > n
func semiperfect(n: int; divs: seq[int]): bool =
if divs.len > 0:
let h = divs[0]
let t = divs[1..^1]
result = if n < h: semiperfect(n, t)
else: n == h or semiperfect(n - h, t) or semiperfect(n, t)
func sieve(limit: int): seq[bool] =
# False denotes abundant and not semi-perfect.
# Only interested in even numbers >= 2.
result.setLen(limit)
for i in countup(2, limit - 1, 2):
if result[i]: continue
let divs = divisors(i)
if not abundant(i, divs):
result[i] = true
elif semiperfect(i, divs):
for j in countup(i, limit - 1, i):
result[j] = true
const Max = 25
let w = sieve(17_000)
var list: seq[int]
echo "The first 25 weird numbers are:" var n = 2 while list.len != Max:
if not w[n]: list.add n inc n, 2
echo list.join(" ")</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
<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, $n;
last if @weird == 25;
}
say "The first 25 weird numbers:\n" . join ' ', @weird;</lang>
- Output:
The first 25 weird numbers: 70 836 4030 5830 7192 7912 9272 10430 10570 10792 10990 11410 11690 12110 12530 12670 13370 13510 13790 13930 14770 15610 15890 16030 16310
Simpler and faster solution:
<lang perl>use 5.010; use strict; use ntheory qw(vecsum divisors divisor_sum);
sub is_pseudoperfect {
my ($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
Phix
Sufficiently fast that I un-optimised it a bit to make it easier to follow. <lang Phix>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})</lang>
- 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
Functional
The first 50 seem to take c. 300 ms
<lang python>Weird numbers
from itertools import chain, count, islice, repeat from functools import reduce from math import sqrt from time import time
- weirds :: Gen [Int]
def weirds():
Non-finite stream of weird numbers.
(Abundant, but not semi-perfect)
OEIS: A006037
def go(n):
ds = descPropDivs(n)
d = sum(ds) - n
return [n] if 0 < d and not hasSum(d, ds) else []
return concatMap(go)(count(1))
- 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))
- concatMap :: (a -> [b]) -> [a] -> [b]
def concatMap(f):
A concatenated list or string over which a function f
has been mapped.
The list monad can be derived by using an (a -> [b])
function which wraps its output in a list (using an
empty list to represent computational failure).
return lambda xs: chain.from_iterable(map(f, xs))
- 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))
)
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 []
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: 284 ms
Racket
<lang racket>#lang racket
(require math/number-theory)
(define (abundant? n proper-divisors)
(> (apply + proper-divisors) n))
(define (semi-perfect? n proper-divisors)
(let recur ((ds proper-divisors) (n n))
(or (zero? n)
(and (positive? n)
(pair? ds)
(or (recur (cdr ds) n)
(recur (cdr ds) (- n (car ds))))))))
(define (weird? n)
(let ((proper-divisors (drop-right (divisors n) 1))) ;; divisors includes n (and (abundant? n proper-divisors) (not (semi-perfect? n proper-divisors)))))
(module+ main
(let recur ((i 0) (n 1) (acc null))
(cond [(= i 25) (reverse acc)]
[(weird? n) (recur (add1 i) (add1 n) (cons n acc))]
[else (recur i (add1 n) acc)])))
(module+ test
(require rackunit) (check-true (weird? 70)) (check-false (weird? 12)))</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)
Raku
(formerly 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
REXX
vanilla version
<lang rexx>/*REXX program finds and displays N weird numbers in a vertical format (with index).*/ parse arg n cols . /*obtain optional arguments from the CL*/ if n== | n=="," then n= 25 /*Not specified? Then use the default.*/ if cols== | cols=="," then cols= 10 /* " " " " " " */ w= 10 /*width of a number in any column. */ if cols>0 then say ' index │'center(' weird numbers', 1 + cols*(w+1) ) if cols>0 then say '───────┼'center("" , 1 + cols*(w+1), '─') idx= 1; $= /*index for the output list; $: 1 line*/ weirds= 0 /*the count of weird numbers (so far).*/
do j=2 by 2 until weirds==n /*examine even integers 'til have 'nuff*/
if \weird(j) then iterate /*Not a weird number? Then skip it. */
weirds= weirds + 1 /*bump the count of weird numbers. */
c= commas(j) /*maybe add commas to the number. */
$= $ right(c, max(w, length(c) ) ) /*add a nice prime ──► list, allow big#*/
if weirds//cols\==0 then iterate /*have we populated a line of output? */
say center(idx, 7)'│' substr($, 2); $= /*display what we have so far (cols). */
idx= idx + cols /*bump the index count for the output*/
end /*j*/
if $\== then say center(idx, 7)"│" substr($, 2) /*possible display residual output.*/ if cols>0 then say '───────┴'center("" , 1 + cols*(w+1), '─') say say 'Found ' commas(weirds) ' weird numbers' exit 0 /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ commas: parse arg _; do ic=length(_)-3 to 1 by -3; _=insert(',', _, ic); end; return _ /*──────────────────────────────────────────────────────────────────────────────────────*/ 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; en
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 and 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 /*zero combo counter; calc. power of 2.*/
do p=1 for 2**#-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.*/</lang>
- output when using the default inputs:
index │ weird numbers ───────┼─────────────────────────────────────────────────────────────────────────────────────────────────────────────── 1 │ 70 836 4,030 5,830 7,192 7,912 9,272 10,430 10,570 10,792 11 │ 10,990 11,410 11,690 12,110 12,530 12,670 13,370 13,510 13,790 13,930 21 │ 14,770 15,610 15,890 16,030 16,310 ───────┴─────────────────────────────────────────────────────────────────────────────────────────────────────────────── Found 25 weird numbers
optimized version
This REXX program was optimized by finding primitive weird numbers (as in the 1st REXX version), and multiplying
them by prime numbers ≥ sigma(primitive weird numbers) to find higher weird numbers.
This version is about 300% faster than the 1st REXX version for larger amount of numbers. <lang rexx>/*REXX program finds and displays N weird numbers in a vertical format (with index).*/ parse arg n cols . /*obtain optional arguments from the CL*/ if n== | n=="," then n= 400 /*Not specified? Then use the default.*/ if cols== | cols=="," then cols= 10 /* " " " " " " */ w= 10 /*width of a number in any column. */ call genP /*generate primes just past Hp. */ if cols>0 then say ' index │'center(' weird numbers', 1 + cols*(w+1) ) if cols>0 then say '───────┼'center("" , 1 + cols*(w+1), '─') @@.= 1; @@.0= 0; @@.2=0 /*semaphores for last decimal digit. */ weirds= 0; !!.= 0 /*the count of weird numbers (so far).*/ idx= 1; $= /*index for the output list; $: 1 line*/
do j=2 by 2 until weirds==n /*examine even integers 'til have 'nuff*/
if \weird(j) then iterate /*Not a weird number? Then skip it. */
weirds= weirds + 1 /*bump the count of weird numbers. */
do a=1 for # until _>hp; if @.a<sigma+j then iterate; _= j*@.a; !!._= 1
end /*a*/
c= commas(j) /*maybe add commas to the number. */
$= $ right(c, max(w, length(c) ) ) /*add a nice prime ──► list, allow big#*/
if weirds//cols\==0 then iterate /*have we populated a line of output? */
say center(idx, 7)'│' substr($, 2); $= /*display what we have so far (cols). */
idx= idx + cols /*bump the index count for the output*/
end /*j*/
if $\== then say center(idx, 7)"│" substr($, 2) /*possible display residual output.*/ if cols>0 then say '───────┴'center("" , 1 + cols*(w+1), '─') say say 'Found ' commas(weirds) ' weird numbers' exit 0 /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ commas: parse arg _; do ic=length(_)-3 to 1 by -3; _=insert(',', _, ic); end; return _ /*──────────────────────────────────────────────────────────────────────────────────────*/ 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 Pdivisors)*/
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). */
/*──────────────────────────────────────────────────────────────────────────────────────*/ genP: hp= 1000 * n /*high Prime limit; define 2 low primes*/
@.1=2; @.2=3; @.3=5; @.4=7; @.5=11 /*define some low primes. */
#=5; s.#= @.# **2 /*number of primes so far; prime². */
/* [↓] generate more primes ≤ high.*/
do j=@.#+2 by 2 for max(0, hp%2-@.#%2-1) /*find odd primes from here on. */
parse var j -1 _; if _==5 then iterate /*J divisible by 5? (right dig)*/
if j// 3==0 then iterate /*" " " 3? */
if j// 7==0 then iterate /*" " " 7? */
/* [↑] the above five lines saves time*/
do k=5 while s.k<=j /* [↓] divide by the known odd primes.*/
if j // @.k == 0 then iterate j /*Is J ÷ X? Then not prime. ___ */
end /*k*/ /* [↑] only process numbers ≤ √ J */
#= #+1; @.#= j; s.#= j*j /*bump # of Ps; assign next P; P²; P# */
end /*j*/; return
/*──────────────────────────────────────────────────────────────────────────────────────*/ weird: procedure expose !!. @@. sigma /*procedure to determine weirdness of X*/
parse arg x -1 _ /*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*/
if x>9e2 then if @@._ then return 0 /*Is an acceptable last decimal digit? */
if !!.x then return 1 /*Is this a prime*previous #? Found one*/
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.*/</lang>
- output when using the default inputs:
index │ weird numbers ───────┼─────────────────────────────────────────────────────────────────────────────────────────────────────────────── 1 │ 70 836 4,030 5,830 7,192 7,912 9,272 10,430 10,570 10,792 11 │ 10,990 11,410 11,690 12,110 12,530 12,670 13,370 13,510 13,790 13,930 21 │ 14,770 15,610 15,890 16,030 16,310 16,730 16,870 17,272 17,570 17,990 31 │ 18,410 18,830 18,970 19,390 19,670 19,810 20,510 21,490 21,770 21,910 41 │ 22,190 23,170 23,590 24,290 24,430 24,710 25,130 25,690 26,110 26,530 51 │ 26,810 27,230 27,790 28,070 28,630 29,330 29,470 30,170 30,310 30,730 61 │ 31,010 31,430 31,990 32,270 32,410 32,690 33,530 34,090 34,370 34,930 71 │ 35,210 35,630 36,470 36,610 37,870 38,290 38,990 39,410 39,830 39,970 81 │ 40,390 41,090 41,510 41,930 42,070 42,490 42,910 43,190 43,330 44,170 91 │ 44,870 45,010 45,290 45,710 46,130 46,270 47,110 47,390 47,810 48,370 101 │ 49,070 49,630 50,330 50,890 51,310 51,730 52,010 52,570 52,990 53,270 111 │ 53,830 54,110 55,090 55,790 56,630 56,770 57,470 57,610 57,890 58,030 121 │ 58,730 59,710 59,990 60,130 60,410 61,390 61,670 61,810 62,090 63,490 131 │ 63,770 64,330 65,030 65,590 65,870 66,290 66,710 67,690 67,970 68,390 141 │ 68,810 69,370 69,790 70,630 70,910 71,330 71,470 72,170 72,310 72,730 151 │ 73,430 73,570 74,270 74,410 74,830 76,090 76,370 76,510 76,790 77,210 161 │ 77,630 78,190 78,610 79,030 80,570 80,710 81,410 81,970 82,670 83,090 171 │ 83,510 84,070 84,910 85,190 85,610 86,030 86,170 86,590 87,430 88,130 181 │ 89,390 89,530 89,810 90,230 90,370 90,790 91,070 91,210 91,490 92,330 191 │ 92,470 92,890 95,270 95,690 96,110 96,670 97,930 98,630 99,610 99,890 201 │ 100,030 100,310 100,730 101,290 101,570 101,710 102,130 102,970 103,670 103,810 211 │ 104,090 104,230 104,510 104,930 105,770 106,610 107,170 108,010 108,430 108,710 221 │ 109,130 109,690 109,970 110,530 110,810 111,790 112,070 112,490 112,630 112,910 231 │ 113,330 113,470 113,890 114,590 115,990 116,410 116,690 116,830 118,510 118,790 241 │ 118,930 119,630 120,470 120,610 121,310 121,870 122,290 122,710 123,130 124,390 251 │ 124,810 125,090 125,230 126,070 126,770 127,610 128,170 129,290 130,270 130,690 261 │ 130,970 131,110 131,390 131,530 132,230 133,070 133,490 133,910 135,170 135,310 271 │ 136,430 136,570 138,110 138,530 139,090 139,510 139,790 139,930 140,210 140,770 281 │ 141,190 141,890 142,030 142,730 143,710 144,410 144,830 145,670 145,810 146,090 291 │ 146,230 146,930 147,770 147,910 149,030 149,170 149,590 149,870 150,010 150,710 301 │ 151,270 152,530 154,210 154,490 154,910 155,470 156,590 156,730 157,010 157,570 311 │ 158,690 158,830 159,110 159,670 160,090 160,510 160,790 161,630 161,770 163,310 321 │ 163,730 163,870 164,290 164,570 164,990 165,970 166,390 166,670 166,810 167,230 331 │ 167,510 167,930 168,770 169,190 169,610 170,590 170,870 171,290 172,130 172,690 341 │ 173,110 173,390 175,210 176,470 177,170 177,730 178,010 178,430 178,570 178,990 351 │ 180,530 181,370 181,510 182,630 183,190 183,470 184,310 185,290 185,990 186,130 361 │ 186,410 186,970 187,390 187,810 188,090 188,230 188,510 188,930 189,490 189,770 371 │ 189,910 190,330 191,030 191,170 191,870 192,430 192,710 193,690 194,390 195,230 381 │ 195,370 195,790 196,070 196,210 197,330 198,310 198,590 199,010 199,570 199,990 391 │ 200,270 201,530 202,090 202,790 203,210 203,630 204,190 204,890 205,730 206,710 ───────┴─────────────────────────────────────────────────────────────────────────────────────────────────────────────── Found 400 weird numbers
Ruby
<lang ruby>def divisors(n)
divs = [1] divs2 = []
i = 2
while i * i <= n
if n % i == 0 then
j = (n / i).to_i
divs.append(i)
if i != j then
divs2.append(j)
end
end
i = i + 1 end
divs2 += divs.reverse return divs2
end
def abundant(n, divs)
return divs.sum > n
end
def semiperfect(n, divs)
if divs.length > 0 then
h = divs[0]
t = divs[1..-1]
if n < h then
return semiperfect(n, t)
else
return n == h || semiperfect(n - h, t) || semiperfect(n, t)
end
else
return false
end
end
def sieve(limit)
w = Array.new(limit, false)
i = 2
while i < limit
if not w[i] then
divs = divisors(i)
if not abundant(i, divs) then
w[i] = true
elsif semiperfect(i, divs) then
j = i
while j < limit
w[j] = true
j = j + i
end
end
end
i = i + 2
end
return w
end
def main
w = sieve(17000)
count = 0
max = 25
print "The first %d weird numbers:\n" % [max]
n = 2
while count < max
if not w[n] then
print n, " "
count = count + 1
end
n = n + 2
end
print "\n"
end
main()</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
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
Visual Basic .NET
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.
Wren
<lang ecmascript>import "/math" for Int, Nums import "/trait" for Stepped
var semiperfect // recursive semiperfect = Fn.new { |n, divs|
var le = divs.count if (le == 0) return false var h = divs[0] if (n == h) return true if (le == 1) return false var t = divs[1..-1] if (n < h) return semiperfect.call(n, t) return semiperfect.call(n-h, t) || semiperfect.call(n, t)
}
var sieve = Fn.new { |limit|
// 'false' denotes abundant and not semi-perfect.
// Only interested in even numbers >= 2
var w = List.filled(limit, false)
for (j in Stepped.new(6...limit, 6)) w[j] = true // eliminate multiples of 3
for (i in Stepped.new(2...limit, 2)) {
if (!w[i]) {
var divs = Int.properDivisors(i)
var sum = Nums.sum(divs)
if (sum <= i) {
w[i] = true
} else if (semiperfect.call(sum-i, divs)) {
for (j in Stepped.new(i...limit, i)) w[j] = true
}
}
}
return w
}
var start = System.clock var limit = 16313 var w = sieve.call(limit) var count = 0 var max = 25 System.print("The first 25 weird numbers are:") var n = 2 while (count < max) {
if (!w[n]) {
System.write("%(n) ")
count = count + 1
}
n = n + 2
} System.print() System.print("\nTook %(((System.clock-start)*1000).round) milliseconds")</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 Took 144 milliseconds
zkl
<lang zkl>fcn properDivs(n){
if(n==1) return(T);
( pd:=[1..(n).toFloat().sqrt()].filter('wrap(x){ n%x==0 }) )
.pump(pd,'wrap(pd){ if(pd!=1 and (y:=n/pd)!=pd ) y else Void.Skip })
} fcn abundant(n,divs){ divs.sum(0) > n } fcn semiperfect(n,divs){
if(divs){
h,t := divs[0], divs[1,*];
if(n<h) return(semiperfect(n,t));
return((n==h) or semiperfect(n - h, t) or semiperfect(n, t));
}
False
} fcn sieve(limit){
// False denotes abundant and not semi-perfect.
// Only interested in even numbers >= 2
w:=List.createLong(limit,False);
foreach i in ([2..limit - 1, 2]){
if(w[i]) continue;
divs:=properDivs(i);
if(not abundant(i,divs)) w[i]=True;
else if(semiperfect(i,divs))
{ foreach j in ([i..limit - 1, i]){ w[j]=True; } }
} w
}</lang> <lang zkl>w,count,max := sieve(17_000), 0, 25; println("The first 25 weird numbers are:"); foreach n in ([2..* ,2]){
if(not w[n]){ print("%d ".fmt(n)); count+=1; }
if(count>=max) break;
} 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