# Self numbers

Self numbers
You are encouraged to solve this task according to the task description, using any language you may know.

A number n is a self number if there is no number g such that g + the sum of g's digits = n. So 18 is not a self number because 9+9=18, 43 is not a self number because 35+5+3=43.

``` Display the first 50 self numbers;
I believe that the 100000000th self number is 1022727208. You should either confirm or dispute my conjecture.
```

224036583-1 is a Mersenne prime, claimed to also be a self number. Extra credit to anyone proving it.

## ALGOL 68

Translation of: Go
```BEGIN # find some self numbers numbers n such that there is no g such that g + sum of g's digits = n #
INT max number = 1 999 999 999 + 82; # maximum n plus g we will condifer #
# sieve the self numbers up to 1 999 999 999 #
[ 0 : max number ]BOOL self; FOR i TO UPB self DO self[ i ] := TRUE OD;
INT n := 0;
FOR s0 FROM 0 TO 1 DO
FOR d1 FROM 0 TO 9 DO
INT s1 = s0 + d1;
FOR d2 FROM 0 TO 9 DO
INT s2 = s1 + d2;
FOR d3 FROM 0 TO 9 DO
INT s3 = s2 + d3;
FOR d4 FROM 0 TO 9 DO
INT s4 = s3 + d4;
FOR d5 FROM 0 TO 9 DO
INT s5 = s4 + d5;
FOR d6 FROM 0 TO 9 DO
INT s6 = s5 + d6;
FOR d7 FROM 0 TO 9 DO
INT s7 = s6 + d7;
FOR d8 FROM 0 TO 9 DO
INT s8 = s7 + d8;
FOR d9 FROM 0 TO 9 DO
INT s9 = s8 + d9;
self[ s9 + n ] := FALSE;
n +:= 1
OD
OD
OD
OD
OD
OD
OD
OD
OD
OD;
# show the first 50 self numbers #
INT s count := 0;
FOR i TO UPB self WHILE s count < 50 DO
IF self[ i ] THEN
print( ( " ", whole( i, -3 ) ) );
IF ( s count +:= 1 ) MOD 18 = 0 THEN print( ( newline ) ) FI
FI
OD;
print( ( newline ) );
# show the self numbers with power-of-10 indxes #
INT s show := 1;
s count    := 0;
print( ( "              nth self", newline ) );
print( ( "        n       number", newline ) );
FOR i TO UPB self DO
IF self[ i ] THEN
s count +:= 1;
IF s count = s show THEN
print( ( whole( s show, -9 ), "  ", whole( i, -11 ), newline ) );
s show *:= 10
FI
FI
OD
END```
Output:
```   1   3   5   7   9  20  31  42  53  64  75  86  97 108 110 121 132 143
154 165 176 187 198 209 211 222 233 244 255 266 277 288 299 310 312 323
334 345 356 367 378 389 400 411 413 424 435 446 457 468
nth self
n       number
1            1
10           64
100          973
1000        10188
10000       102225
100000      1022675
1000000     10227221
10000000    102272662
100000000   1022727208
```

## AppleScript

I couldn't follow the math in the Wikipedia entry, nor the discussion and code here so far. But an initial expedient of generating a list of all the integers from 1 to just over ten times the required number of results and then deleting those that could be derived by the described method revealed the sequencing pattern on which the code below is based. On the test machine, it completes all three of the tests at the bottom in a total of around a millisecond.

```(*
Base-10 self numbers by index (single or range).
Follows an observed sequence pattern whereby, after the initial single-digit odd numbers, self numbers are
grouped in runs whose members occur at numeric intervals of 11. Runs after the first one come in blocks of
ten: eight runs of ten numbers followed by two shorter runs. The numeric interval between runs is usually 2,
but that between shorter runs, and their length, depend on the highest-order digit change occurring in them.
This connection with significant digit change means every ten blocks form a higher-order block, every ten
of these a higher-order-still block, and so on.

The code below appears to be good up to the last self number before 10^12 — ie. 999,999,999,997, which is
returned as the 97,777,777,792nd such number. After this, instead of zero-length shorter runs, the actual
pattern apparently starts again with a single run of 10, like the one at the beginning.
*)
on selfNumbers(indexRange)
set indexRange to indexRange as list
-- Script object with subhandlers and associated properties.
script |subscript|
property startIndex : beginning of indexRange
property endIndex : end of indexRange
property counter : 0
property currentSelf : -1
property output : {}

-- Advance to the next self number in the sequence, append it to the output if required, indicate if finished.
set currentSelf to currentSelf + interval
set counter to counter + 1
if (counter < startIndex) then return false
set end of my output to currentSelf
return (counter = endIndex)

-- If necessary, fast forward to the last self number before the lowest-order block containing the first number required.
on fastForward()
if (counter ≥ startIndex) then return
-- The highest-order blocks whose ends this script handles correctly contain 9,777,777,778 self numbers.
-- The difference between equivalently positioned numbers in these blocks is 100,000,000,001.
-- The figures for successively lower-order blocks have successively fewer 7s and 0s!
set indexDiff to 9.777777778E+9
set numericDiff to 1.00000000001E+11
repeat until ((indexDiff < 98) or (counter = startIndex))
set test to counter + indexDiff
if (test < startIndex) then
set counter to test
set currentSelf to (currentSelf + numericDiff)
else
set indexDiff to (indexDiff + 2) div 10
set numericDiff to numericDiff div 10 + 1
end if
end repeat
end fastForward

-- Work out a shorter run length based on the most significant digit change about to happen.
on getShorterRunLength()
set shorterRunLength to 9
set temp to (|subscript|'s currentSelf) div 1000
repeat while (temp mod 10 is 9)
set shorterRunLength to shorterRunLength - 1
set temp to temp div 10
end repeat
return shorterRunLength
end getShorterRunLength
end script

-- Main process. Start with the single-digit odd numbers and first run.
repeat 5 times
if (|subscript|'s doneAfterAdding(2)) then return |subscript|'s output
end repeat
repeat 9 times
if (|subscript|'s doneAfterAdding(11)) then return |subscript|'s output
end repeat
-- Fast forward if the start index hasn't yet been reached.
tell |subscript| to fastForward()

-- Sequencing loop, per lowest-order block.
repeat
-- Eight ten-number runs, each at a numeric interval of 2 from the end of the previous one.
repeat 8 times
if (|subscript|'s doneAfterAdding(2)) then return |subscript|'s output
repeat 9 times
if (|subscript|'s doneAfterAdding(11)) then return |subscript|'s output
end repeat
end repeat
-- Two shorter runs, the second at an interval inversely related to their length.
set shorterRunLength to |subscript|'s getShorterRunLength()
repeat with interval in {2, 2 + (10 - shorterRunLength) * 13}
if (|subscript|'s doneAfterAdding(interval)) then return |subscript|'s output
repeat (shorterRunLength - 1) times
if (|subscript|'s doneAfterAdding(11)) then return |subscript|'s output
end repeat
end repeat
end repeat
end selfNumbers

-- Demo calls:
-- First to fiftieth self numbers.
selfNumbers({1, 50})
--> {1, 3, 5, 7, 9, 20, 31, 42, 53, 64, 75, 86, 97, 108, 110, 121, 132, 143, 154, 165, 176, 187, 198, 209, 211, 222, 233, 244, 255, 266, 277, 288, 299, 310, 312, 323, 334, 345, 356, 367, 378, 389, 400, 411, 413, 424, 435, 446, 457, 468}

-- One hundred millionth:
selfNumbers(100000000)
--> {1.022727208E+9}

-- 97,777,777,792nd:
selfNumbers(9.7777777792E+10)
--> {9.99999999997E+11}
```

## AWK

```# syntax: GAWK -f SELF_NUMBERS.AWK
# converted from Go (low memory example)
BEGIN {
print("HH:MM:SS      INDEX       SELF")
print("-------- ---------- ----------")
count = 0
digits = 1
i = 1
last_self = 0
offset = 9
pow = 10
while (count < 1E8) {
is_self = 1
start = max(i-offset,0)
sum = sum_digits(start)
for (j=start; j<i; j++) {
if (j + sum == i) {
is_self = 0
break
}
sum = ((j+1) % 10 != 0) ? ++sum : sum_digits(j+1)
}
if (is_self) {
last_self = i
if (++count <= 50) {
selfs = selfs i " "
}
}
if (++i % pow == 0) {
pow *= 10
digits++
offset = digits * 9
}
if (count ~ /^10*\$/ && arr[count]++ == 0) {
printf("%8s %10s %10s\n",strftime("%H:%M:%S"),count,last_self)
}
}
printf("\nfirst 50 self numbers:\n%s\n",selfs)
exit(0)
}
function sum_digits(x,  sum,y) {
while (x) {
y = x % 10
sum += y
x = int(x/10)
}
return(sum)
}
function max(x,y) { return((x > y) ? x : y) }
```
Output:
```HH:MM:SS      INDEX       SELF
-------- ---------- ----------
00:36:35          1          1
00:36:35         10         64
00:36:35        100        973
00:36:35       1000      10188
00:36:36      10000     102225
00:36:46     100000    1022675
00:38:49    1000000   10227221
01:03:01   10000000  102272662
05:27:35  100000000 1022727208

first 50 self numbers:
1 3 5 7 9 20 31 42 53 64 75 86 97 108 110 121 132 143 154 165 176 187 198 209 211 222 233 244 255 266 277 288 299 310 312 323 334 345 356 367 378 389 400 411 413 424 435 446 457 468
```

## C

### Sieve based

Translation of: Go

About 25% faster than Go (using GCC 7.5.0 -O3) mainly due to being able to iterate through the sieve using a pointer.

```#include <stdio.h>
#include <stdlib.h>
#include <time.h>

typedef unsigned char bool;

#define TRUE 1
#define FALSE 0
#define MILLION 1000000
#define BILLION 1000 * MILLION
#define MAX_COUNT 2*BILLION + 9*9 + 1

void sieve(bool *sv) {
int n = 0, s[8], a, b, c, d, e, f, g, h, i, j;
for (a = 0; a < 2; ++a) {
for (b = 0; b < 10; ++b) {
s[0] = a + b;
for (c = 0; c < 10; ++c) {
s[1] = s[0] + c;
for (d = 0; d < 10; ++d) {
s[2] = s[1] + d;
for (e = 0; e < 10; ++e) {
s[3] = s[2] + e;
for (f = 0; f < 10; ++f) {
s[4] = s[3] + f;
for (g = 0; g < 10; ++g) {
s[5] = s[4] + g;
for (h = 0; h < 10; ++h) {
s[6] = s[5] + h;
for (i = 0; i < 10; ++i) {
s[7] = s[6] + i;
for (j = 0; j < 10; ++j) {
sv[s[7] + j+ n++] = TRUE;
}
}
}
}
}
}
}
}
}
}
}

int main() {
int count = 0;
clock_t begin = clock();
bool *p, *sv = (bool*) calloc(MAX_COUNT, sizeof(bool));
sieve(sv);
printf("The first 50 self numbers are:\n");
for (p = sv; p < sv + MAX_COUNT; ++p) {
if (!*p) {
if (++count <= 50) printf("%ld ", p-sv);
if (count == 100 * MILLION) {
printf("\n\nThe 100 millionth self number is %ld\n", p-sv);
break;
}
}
}
free(sv);
printf("Took %lf seconds.\n", (double)(clock() - begin) / CLOCKS_PER_SEC);
return 0;
}
```
Output:
```The first 50 self numbers are:
1 3 5 7 9 20 31 42 53 64 75 86 97 108 110 121 132 143 154 165 176 187 198 209 211 222 233 244 255 266 277 288 299 310 312 323 334 345 356 367 378 389 400 411 413 424 435 446 457 468

The 100 millionth self number is 1022727208
Took 1.521486 seconds.
```

### Extended

Translation of: Pascal
```#include <stdio.h>
#include <stdlib.h>
#include <time.h>

typedef unsigned char bool;

#define TRUE 1
#define FALSE 0
#define MILLION 1000000LL
#define BILLION 1000 * MILLION
#define MAX_COUNT 103LL*10000*10000 + 11*9 + 1

int digitSum[10000];

void init() {
int i = 9999, s, t, a, b, c, d;
for (a = 9; a >= 0; --a) {
for (b = 9; b >= 0; --b) {
s = a + b;
for (c = 9; c >= 0; --c) {
t = s + c;
for (d = 9; d >= 0; --d) {
digitSum[i] = t + d;
--i;
}
}
}
}
}

void sieve(bool *sv) {
int a, b, c;
long long s, n = 0;
for (a = 0; a < 103; ++a) {
for (b = 0; b < 10000; ++b) {
s = digitSum[a] + digitSum[b] + n;
for (c = 0; c < 10000; ++c) {
sv[digitSum[c]+s] = TRUE;
++s;
}
n += 10000;
}
}
}

int main() {
long long count = 0, limit = 1;
clock_t begin = clock(), end;
bool *p, *sv = (bool*) calloc(MAX_COUNT, sizeof(bool));
init();
sieve(sv);
printf("Sieving took %lf seconds.\n", (double)(clock() - begin) / CLOCKS_PER_SEC);
printf("\nThe first 50 self numbers are:\n");
for (p = sv; p < sv + MAX_COUNT; ++p) {
if (!*p) {
if (++count <= 50) {
printf("%ld ", p-sv);
} else {
printf("\n\n     Index  Self number\n");
break;
}
}
}
count = 0;
for (p = sv; p < sv + MAX_COUNT; ++p) {
if (!*p) {
if (++count == limit) {
printf("%10lld  %11ld\n", count, p-sv);
limit *= 10;
if (limit == 10 * BILLION) break;
}
}
}
free(sv);
printf("\nOverall took %lf seconds.\n", (double)(clock() - begin) / CLOCKS_PER_SEC);
return 0;
}
```
Output:
```Sieving took 7.429969 seconds.

The first 50 self numbers are:
1 3 5 7 9 20 31 42 53 64 75 86 97 108 110 121 132 143 154 165 176 187 198 209 211 222 233 244 255 266 277 288 299 310 312 323 334 345 356 367 378 389 400 411 413 424 435 446 457 468

Index  Self number
1            1
10           64
100          973
1000        10188
10000       102225
100000      1022675
1000000     10227221
10000000    102272662
100000000   1022727208
1000000000  10227272649

Overall took 11.574158 seconds.
```

## C++

Translation of: Java
```#include <array>
#include <iomanip>
#include <iostream>

const int MC = 103 * 1000 * 10000 + 11 * 9 + 1;
std::array<bool, MC + 1> SV;

void sieve() {
std::array<int, 10000> dS;
for (int a = 9, i = 9999; a >= 0; a--) {
for (int b = 9; b >= 0; b--) {
for (int c = 9, s = a + b; c >= 0; c--) {
for (int d = 9, t = s + c; d >= 0; d--) {
dS[i--] = t + d;
}
}
}
}
for (int a = 0, n = 0; a < 103; a++) {
for (int b = 0, d = dS[a]; b < 1000; b++, n += 10000) {
for (int c = 0, s = d + dS[b] + n; c < 10000; c++) {
SV[dS[c] + s++] = true;
}
}
}
}

int main() {
sieve();

std::cout << "The first 50 self numbers are:\n";
for (int i = 0, count = 0; count <= 50; i++) {
if (!SV[i]) {
count++;
if (count <= 50) {
std::cout << i << ' ';
} else {
std::cout << "\n\n       Index     Self number\n";
}
}
}
for (int i = 0, limit = 1, count = 0; i < MC; i++) {
if (!SV[i]) {
if (++count == limit) {
//System.out.printf("%,12d   %,13d%n", count, i);
std::cout << std::setw(12) << count << "   " << std::setw(13) << i << '\n';
limit *= 10;
}
}
}

return 0;
}
```
Output:
```The first 50 self numbers are:
1 3 5 7 9 20 31 42 53 64 75 86 97 108 110 121 132 143 154 165 176 187 198 209 211 222 233 244 255 266 277 288 299 310 312 323 334 345 356 367 378 389 400 411 413 424 435 446 457 468

Index     Self number
1               1
10              64
100             973
1000           10188
10000          102225
100000         1022675
1000000        10227221
10000000       102272662
100000000      1022727208```

## C#

Translation of: Pascal

via

Translation of: Go

(third version) Stripped down, as C# limits the size of an array to Int32.MaxValue, so the sieve isn't large enough to hit the 1,000,000,000th value.

```using System;
using static System.Console;

class Program {

const int mc = 103 * 1000 * 10000 + 11 * 9 + 1;

static bool[] sv = new bool[mc + 1];

static void sieve() { int[] dS = new int[10000];
for (int a = 9, i = 9999; a >= 0; a--)
for (int b = 9; b >= 0; b--)
for (int c = 9, s = a + b; c >= 0; c--)
for (int d = 9, t = s + c; d >= 0; d--)
dS[i--] = t + d;
for (int a = 0, n = 0; a < 103; a++)
for (int b = 0, d = dS[a]; b < 1000; b++, n += 10000)
for (int c = 0, s = d + dS[b] + n; c < 10000; c++)
sv[dS[c] + s++] = true; }

static void Main() { DateTime st = DateTime.Now; sieve();
WriteLine("Sieving took {0}s", (DateTime.Now - st).TotalSeconds);
WriteLine("\nThe first 50 self numbers are:");
for (int i = 0, count = 0; count <= 50; i++) if (!sv[i]) {
count++; if (count <= 50) Write("{0} ", i);
else WriteLine("\n\n       Index     Self number"); }
for (int i = 0, limit = 1, count = 0; i < mc; i++)
if (!sv[i]) if (++count == limit) {
WriteLine("{0,12:n0}   {1,13:n0}", count, i);
if (limit == 1e9) break; limit *= 10; }
WriteLine("\nOverall took {0}s", (DateTime.Now - st). TotalSeconds);
}
}
```
Output:

Timing from tio.run

```Sieving took 3.4266187s

The first 50 self numbers are:
1 3 5 7 9 20 31 42 53 64 75 86 97 108 110 121 132 143 154 165 176 187 198 209 211 222 233 244 255 266 277 288 299 310 312 323 334 345 356 367 378 389 400 411 413 424 435 446 457 468

Index     Self number
1               1
10              64
100             973
1,000          10,188
10,000         102,225
100,000       1,022,675
1,000,000      10,227,221
10,000,000     102,272,662
100,000,000   1,022,727,208

Overall took 7.0237244s```

## EasyLang

Translation of: Go
```fastfunc digsum h .
while h > 0
sum += h mod 10
h = h div 10
.
return sum
.
fastfunc isself start i .
j = start
sum = digsum start
while j < i
if j + sum = i
return 0
.
sum += 1
j += 1
if j mod 10 = 0
sum = digsum j
.
.
return 1
.
proc main . .
i = 1
po = 10
digits = 1
offs = 9
repeat
start = higher (i - offs) 0
if isself start i = 1
cnt += 1
if cnt <= 50
write i & " "
.
.
until cnt = 100000000
i += 1
if i mod po = 0
po *= 10
digits += 1
offs = digits * 9
.
.
print ""
print i
.
main```
Output:
```1 3 5 7 9 20 31 42 53 64 75 86 97 108 110 121 132 143 154 165 176 187 198 209 211 222 233 244 255 266 277 288 299 310 312 323 334 345 356 367 378 389 400 411 413 424 435 446 457 468
1022727208
```

## Elixir

```defmodule SelfNums do

def digAndSum(number) when is_number(number) do
Integer.digits(number) |>
Enum.reduce( 0, fn(num, acc) -> num + acc end ) |>
(fn(x) -> x + number end).()
end

def selfFilter(list, firstNth) do
numRange = Enum.to_list 1..firstNth
numRange -- list
end

end

defmodule SelfTest do

import SelfNums
stop = 1000
Enum.to_list 1..stop |>
Enum.map(&digAndSum/1) |>
SelfNums.selfFilter(stop) |>
Enum.take(50) |>
IO.inspect

end
```
Output:
```[1, 3, 5, 7, 9, 20, 31, 42, 53, 64, 75, 86, 97, 108, 110, 121, 132, 143, 154,
165, 176, 187, 198, 209, 211, 222, 233, 244, 255, 266, 277, 288, 299, 310, 312,
323, 334, 345, 356, 367, 378, 389, 400, 411, 413, 424, 435, 446, 457, 468]
```

## F#

```// Self numbers. Nigel Galloway: October 6th., 2020
let fN g=let rec fG n g=match n/10 with 0->n+g |i->fG i (g+(n%10)) in fG g g
let Self=let rec Self n i g=seq{let g=g@([n..i]|>List.map fN) in yield! List.except g [n..i]; yield! Self (n+100) (i+100) (List.filter(fun n->n>i) g)} in Self 0 99 []

Self |> Seq.take 50 |> Seq.iter(printf "%d "); printfn ""
printfn "\n%d" (Seq.item 99999999 Self)
```
Output:
```1 3 5 7 9 20 31 42 53 64 75 86 97 108 110 121 132 143 154 165 176 187 198 209 211 222 233 244 255 266 277 288 299 310 312 323 334 345 356 367 378 389 400 411 413 424 435 446 457 468

1022727208
```

## FreeBASIC

Translation of: Ring
```Print "The first 50 self numbers are:"
Dim As Boolean flag
Dim As Ulong m, p, sum, number(), sum2
Dim As Ulong n = 0
Dim As Ulong num = 0
Dim As Ulong limit = 51
Dim As Ulong limit2 = 100000000
Dim As String strnum

Do
n += 1
For m = 1 To n
flag = True
sum = 0
strnum = Str(m)
For p = 1 To Len(strnum)
sum += Val(Mid(strnum,p,1))
Next p
sum2 = m + sum
If sum2 = n Then
flag = False
Exit For
Else
flag = True
End If
Next m
If flag = True Then
num += 1
If num < limit Then Print ""; num; ". "; n
If num >= limit2 Then
Print "The "; limit2; "th self number is: "; n
Exit Do
End If
End If
Loop
Sleep```

## Go

### Low memory

Simple-minded, uses very little memory (no sieve) but slow - over 2.5 minutes.

```package main

import (
"fmt"
"time"
)

func sumDigits(n int) int {
sum := 0
for n > 0 {
sum += n % 10
n /= 10
}
return sum
}

func max(x, y int) int {
if x > y {
return x
}
return y
}

func main() {
st := time.Now()
count := 0
var selfs []int
i := 1
pow := 10
digits := 1
offset := 9
lastSelf := 0
for count < 1e8 {
isSelf := true
start := max(i-offset, 0)
sum := sumDigits(start)
for j := start; j < i; j++ {
if j+sum == i {
isSelf = false
break
}
if (j+1)%10 != 0 {
sum++
} else {
sum = sumDigits(j + 1)
}
}
if isSelf {
count++
lastSelf = i
if count <= 50 {
selfs = append(selfs, i)
if count == 50 {
fmt.Println("The first 50 self numbers are:")
fmt.Println(selfs)
}
}
}
i++
if i%pow == 0 {
pow *= 10
digits++
offset = digits * 9
}
}
fmt.Println("\nThe 100 millionth self number is", lastSelf)
fmt.Println("Took", time.Since(st))
}
```
Output:
```The first 50 self numbers are:
[1 3 5 7 9 20 31 42 53 64 75 86 97 108 110 121 132 143 154 165 176 187 198 209 211 222 233 244 255 266 277 288 299 310 312 323 334 345 356 367 378 389 400 411 413 424 435 446 457 468]

The 100 millionth self number is 1022727208
Took 2m35.531949399s
```

### Sieve based

Simple sieve, requires a lot of memory but quick - around 2 seconds.

Nested 'for's used rather than a recursive function for extra speed.

Have also incorporated Enter your username's suggestion (see Talk page) of using partial sums for each loop which improves performance by about 25%.

```package main

import (
"fmt"
"time"
)

func sieve() []bool {
sv := make([]bool, 2*1e9+9*9 + 1)
n := 0
var s [8]int
for a := 0; a < 2; a++ {
for b := 0; b < 10; b++ {
s[0] = a + b
for c := 0; c < 10; c++ {
s[1] = s[0] + c
for d := 0; d < 10; d++ {
s[2] = s[1] + d
for e := 0; e < 10; e++ {
s[3] = s[2] + e
for f := 0; f < 10; f++ {
s[4] = s[3] + f
for g := 0; g < 10; g++ {
s[5] = s[4] + g
for h := 0; h < 10; h++ {
s[6] = s[5] + h
for i := 0; i < 10; i++ {
s[7] = s[6] + i
for j := 0; j < 10; j++ {
sv[s[7]+j+n] = true
n++
}
}
}
}
}
}
}
}
}
}
return sv
}

func main() {
st := time.Now()
sv := sieve()
count := 0
fmt.Println("The first 50 self numbers are:")
for i := 0; i < len(sv); i++ {
if !sv[i] {
count++
if count <= 50 {
fmt.Printf("%d ", i)
}
if count == 1e8 {
fmt.Println("\n\nThe 100 millionth self number is", i)
break
}
}
}
fmt.Println("Took", time.Since(st))
}
```
Output:
```The first 50 self numbers are:
1 3 5 7 9 20 31 42 53 64 75 86 97 108 110 121 132 143 154 165 176 187 198 209 211 222 233 244 255 266 277 288 299 310 312 323 334 345 356 367 378 389 400 411 413 424 435 446 457 468

The 100 millionth self number is 1022727208
Took 1.984969034s
```

### Extended

Translation of: Pascal

This uses horst.h's ideas (see Talk page) to find up to the 1 billionth self number in a reasonable time and using less memory than the simple 'sieve based' approach above would have needed.

```package main

import (
"fmt"
"time"
)

const MAX_COUNT = 103*1e4*1e4 + 11*9 + 1

var sv = make([]bool, MAX_COUNT+1)
var digitSum = make([]int, 1e4)

func init() {
i := 9999
var s, t int
for a := 9; a >= 0; a-- {
for b := 9; b >= 0; b-- {
s = a + b
for c := 9; c >= 0; c-- {
t = s + c
for d := 9; d >= 0; d-- {
digitSum[i] = t + d
i--
}
}
}
}
}

func sieve() {
n := 0
for a := 0; a < 103; a++ {
for b := 0; b < 1e4; b++ {
s := digitSum[a] + digitSum[b] + n
for c := 0; c < 1e4; c++ {
sv[digitSum[c]+s] = true
s++
}
n += 1e4
}
}
}

func main() {
st := time.Now()
sieve()
fmt.Println("Sieving took", time.Since(st))
count := 0
fmt.Println("\nThe first 50 self numbers are:")
for i := 0; i < len(sv); i++ {
if !sv[i] {
count++
if count <= 50 {
fmt.Printf("%d ", i)
} else {
fmt.Println("\n\n     Index  Self number")
break
}
}
}
count = 0
limit := 1
for i := 0; i < len(sv); i++ {
if !sv[i] {
count++
if count == limit {
fmt.Printf("%10d  %11d\n", count, i)
limit *= 10
if limit == 1e10 {
break
}
}
}
}
fmt.Println("\nOverall took", time.Since(st))
}
```
Output:
```Sieving took 8.286841692s

The first 50 self numbers are:
1 3 5 7 9 20 31 42 53 64 75 86 97 108 110 121 132 143 154 165 176 187 198 209 211 222 233 244 255 266 277 288 299 310 312 323 334 345 356 367 378 389 400 411 413 424 435 446 457 468

Index  Self number
1            1
10           64
100          973
1000        10188
10000       102225
100000      1022675
1000000     10227221
10000000    102272662
100000000   1022727208
1000000000  10227272649

Overall took 14.647314803s
```

The solution is quite straightforward. The length of the foreseeing window in filtering procedure (81) is chosen empirically and doesn't have any theoretical background.

```import Control.Monad (forM_)
import Text.Printf

selfs :: [Integer]
selfs = sieve (sumFs [0..]) [0..]
where
sumFs = zipWith (+) [ a+b+c+d+e+f+g+h+i+j
| a <- [0..9] , b <- [0..9]
, c <- [0..9] , d <- [0..9]
, e <- [0..9] , f <- [0..9]
, g <- [0..9] , h <- [0..9]
, i <- [0..9] , j <- [0..9] ]

-- More idiomatic list generator is about three times slower
--  sumFs = zipWith (+) \$ sum <\$> replicateM  10 [0..9]

sieve (f:fs) (n:ns)
| n > f = sieve fs (n:ns)
| n `notElem` take 81 (f:fs) = n : sieve (f:fs) ns
| otherwise = sieve (f:fs) ns

main = do
print \$ take 50 selfs
forM_ [1..8] \$ \i ->
printf "1e%v\t%v\n" (i :: Int) (selfs !! (10^i-1))
```
```\$ ghc -O2 SelfNum.hs && time ./SelfNum
[1,3,5,7,9,20,31,42,53,64,75,86,97,108,110,121,132,143,154,165,176,187,198,209,211,222,233,244,255,266,277,288,299,310,312,323,334,345,356,367,378,389,400,411,413,424,435,446,457,468]
1e1	64
1e2	973
1e3	10188
1e4	102225
1e5	1022675
1e6	10227221
1e7	102272662
1e8	1022727208
275.98 user 3.11 system 4:41.02 elapsed```

## Java

Translation of: C#
```public class SelfNumbers {
private static final int MC = 103 * 1000 * 10000 + 11 * 9 + 1;
private static final boolean[] SV = new boolean[MC + 1];

private static void sieve() {
int[] dS = new int[10_000];
for (int a = 9, i = 9999; a >= 0; a--) {
for (int b = 9; b >= 0; b--) {
for (int c = 9, s = a + b; c >= 0; c--) {
for (int d = 9, t = s + c; d >= 0; d--) {
dS[i--] = t + d;
}
}
}
}
for (int a = 0, n = 0; a < 103; a++) {
for (int b = 0, d = dS[a]; b < 1000; b++, n += 10000) {
for (int c = 0, s = d + dS[b] + n; c < 10000; c++) {
SV[dS[c] + s++] = true;
}
}
}
}

public static void main(String[] args) {
sieve();
System.out.println("The first 50 self numbers are:");
for (int i = 0, count = 0; count <= 50; i++) {
if (!SV[i]) {
count++;
if (count <= 50) {
System.out.printf("%d ", i);
} else {
System.out.printf("%n%n       Index     Self number%n");
}
}
}
for (int i = 0, limit = 1, count = 0; i < MC; i++) {
if (!SV[i]) {
if (++count == limit) {
System.out.printf("%,12d   %,13d%n", count, i);
limit *= 10;
}
}
}
}
}
```
Output:
```The first 50 self numbers are:
1 3 5 7 9 20 31 42 53 64 75 86 97 108 110 121 132 143 154 165 176 187 198 209 211 222 233 244 255 266 277 288 299 310 312 323 334 345 356 367 378 389 400 411 413 424 435 446 457 468

Index     Self number
1               1
10              64
100             973
1,000          10,188
10,000         102,225
100,000       1,022,675
1,000,000      10,227,221
10,000,000     102,272,662
100,000,000   1,022,727,208```

## jq

Works with: jq
```def sumdigits: tostring | explode | map([.]|implode|tonumber) | add;

def gsum: . + sumdigits;

def isnonself:
def ndigits: tostring|length;
. as \$i
| (\$i - (\$i|ndigits)*9) as \$n
| any( range(\$i-1; [0,\$n]|max; -1);
gsum == \$i);

# an array
def last81:
[limit(81; range(1; infinite) | select(isnonself))];

# output an unbounded stream
def selfnumbers:
foreach range(1; infinite) as \$i ( [0, last81];
.[0] = false
| .[1] as \$last81
| if \$last81 | bsearch(\$i) < 0
then .[0] = \$i
| if \$i > \$last81[-1] then .[1] = \$last81[1:] + [\$i | gsum ] else . end
else .
end;
.[0] | select(.) );

"The first 50 self numbers are:", last81[:50],
"",
(nth(100000000 - 1;  selfnumbers)
| if . == 1022727208
then "Yes, \(.) is the 100,000,000th self number."
else "No,  \(.i) is actually the 100,000,000th self number."
end)```
Output:
```The first 50 self numbers are:
[2,4,6,8,10,11,12,13,14,15,16,17,18,19,21,22,23,24,25,26,27,28,29,30,32,33,34,35,36,37,38,39,40,41,43,44,45,46,47,48,49,50,51,52,54,55,56,57,58,59]

Yes, 1022727208 is the 100,000,000th self number.

```

## Julia

The code first bootstraps a sliding window of size 81 and then uses this as a sieve. Note that 81 is the window size because the sum of digits of 999,999,999 (the largest digit sum of a counting number less than 1022727208) is 81.

```gsum(i) = sum(digits(i)) + i
isnonself(i) = any(x -> gsum(x) == i, i-1:-1:i-max(1, ndigits(i)*9))
const last81 = filter(isnonself, 1:5000)[1:81]

function checkselfnumbers()
i, selfcount = 1, 0
while selfcount <= 100_000_000 && i <= 1022727208
if !(i in last81)
selfcount += 1
if selfcount < 51
print(i, " ")
elseif selfcount == 51
println()
elseif selfcount == 100_000_000
println(i == 1022727208 ?
"Yes, \$i is the 100,000,000th self number." :
"No, instead \$i is the 100,000,000th self number.")
end
end
popfirst!(last81)
push!(last81, gsum(i))
i += 1
end
end

checkselfnumbers()
```
Output:
```1 3 5 7 9 20 31 42 53 64 75 86 97 108 110 121 132 143 154 165 176 187 198 209 211 222 233 244 255 266 277 288 299 310 312 323 334 345 356 367 378 389 400 411 413 424 435 446 457 468
Yes, 1022727208 is the 100,000,000th self number.
```

### Faster version

Translation of: Pascal

Contains tweaks peculiar to the "10 to the nth" self number. Timings include compilation times.

```const MAXCOUNT = 103 * 10000 * 10000 + 11 * 9 + 1

function dosieve!(sieve, digitsum9999)
n = 1
for a in 1:103, b in 1:10000
s = digitsum9999[a] + digitsum9999[b] + n
for c in 1:10000
sieve[digitsum9999[c] + s] = true
s += 1
end
n += 10000
end
end

initdigitsum() = reverse!(vec([sum(k) for k in Iterators.product(9:-1:0, 9:-1:0, 9:-1:0, 9:-1:0)]))

function findselves()
sieve = zeros(Bool, MAXCOUNT+1)
println("Sieve time:")
@time begin
digitsum = initdigitsum()
dosieve!(sieve, digitsum)
end
cnt = 1
for i in 1:MAXCOUNT+1
if !sieve[i]
cnt > 50 && break
print(i, " ")
cnt += 1
end
end
println()
limit, cnt = 1, 0
for i in 0:MAXCOUNT
cnt += 1 - sieve[i + 1]
if cnt == limit
limit *= 10
end
end
end

@time findselves()
```
Output:
```Sieve time:
7.187635 seconds (2 allocations: 78.203 KiB)
1 3 5 7 9 20 31 42 53 64 75 86 97 108 110 121 132 143 154 165 176 187 198 209 211 222 233 244 255 266 277 288 299 310 312 323 334 345 356 367 378 389 400 411 413 424 435 446 457 468
1           1
10          64
100         973
1000       10188
10000      102225
100000     1022675
1000000    10227221
10000000   102272662
100000000  1022727208
1000000000 10227272649
16.999383 seconds (42.92 k allocations: 9.595 GiB, 0.01% gc time)
```

## Kotlin

Translation of: Java
```private const val MC = 103 * 1000 * 10000 + 11 * 9 + 1
private val SV = BooleanArray(MC + 1)

private fun sieve() {
val dS = IntArray(10000)
run {
var a = 9
var i = 9999
while (a >= 0) {
for (b in 9 downTo 0) {
var c = 9
val s = a + b
while (c >= 0) {
var d = 9
val t = s + c
while (d >= 0) {
dS[i--] = t + d
d--
}
c--
}
}
a--
}
}
var a = 0
var n = 0
while (a < 103) {
var b = 0
val d = dS[a]
while (b < 1000) {
var c = 0
var s = d + dS[b] + n
while (c < 10000) {
SV[dS[c] + s++] = true
c++
}
b++
n += 10000
}
a++
}
}

fun main() {
sieve()
println("The first 50 self numbers are:")
run {
var i = 0
var count = 0
while (count <= 50) {
if (!SV[i]) {
count++
if (count <= 50) {
print("\$i ")
} else {
println()
println()
println("       Index     Self number")
}
}
i++
}
}
var i = 0
var limit = 1
var count = 0
while (i < MC) {
if (!SV[i]) {
if (++count == limit) {
println("%,12d   %,13d".format(count, i))
limit *= 10
}
}
i++
}
}
```
Output:
```The first 50 self numbers are:
1 3 5 7 9 20 31 42 53 64 75 86 97 108 110 121 132 143 154 165 176 187 198 209 211 222 233 244 255 266 277 288 299 310 312 323 334 345 356 367 378 389 400 411 413 424 435 446 457 468

Index     Self number
1               1
10              64
100             973
1,000          10,188
10,000         102,225
100,000       1,022,675
1,000,000      10,227,221
10,000,000     102,272,662
100,000,000   1,022,727,208```

## Mathematica/Wolfram Language

```sum[g_] := g + Total@IntegerDigits@g

ming[n_] := n - IntegerLength[n]*9

self[n_] := NoneTrue [Range[ming[n], n - 1], sum[#] == n &]

Module[{t = 1, x = 1},
Reap[
While[t <= 50,
If[self[x], Sow[x]; t++]; x++]
][[2, 1]]]
```
Output:
`{1,3,5,7,9,20,31,42,53,64,75,86,97,108,110,121,132,143,154,165,176,187,198,209,211,222,233,244,255,266,277,288,299,310,312,323,334,345,356,367,378,389,400,411,413,424,435,446,457,468}`

## Nim

In order to use less memory, we have chosen to use indexing at bit level. So, our sieve is a custom object defined by its length in bits and its value which is a sequence of bytes. With bit indexing, the sieve uses eight times less memory than with byte indexing.

Of course, there is a trade off to this strategy: reading values from and writing values to the sieve are significantly slower.

### Simple sieve

Translation of: Go

We use the Go algorithm with bit indexing. As a consequence the sieve uses about 250MB instead of 1 GB. And the program is about five times slower.

Note that we used a sequence of ten nested loops as in the Go solution but we have not memorized the intermediate sums as the C compiler does a good job to detect the loop invariants (remember, Nim produces C code and this code has proved to be quite optimizable by the C compiler). The ten loops looks a lot better this way, too 🙂.

```import bitops, strutils, std/monotimes, times

const MaxCount = 2 * 1_000_000_000 + 9 * 9

# Bit string used to represent an array of booleans.
type BitString = object
len: Natural        # length in bits.
values: seq[byte]   # Sequence containing the bits.

proc newBitString(n: Natural): BitString =
## Return a new bit string of length "n" bits.
result.len = n
result.values.setLen((n + 7) shr 3)

template checkIndex(i, length: Natural) {.used.} =
## Check if index "i" is less than the array length.
if i >= length:
raise newException(IndexDefect, "index \$1 not in 0 .. \$2".format(i, length))

proc `[]`(bs: BitString; i: Natural): bool =
## Return the value of bit at index "i" as a boolean.
when compileOption("boundchecks"):
checkIndex(i, bs.len)
result = bs.values[i shr 3].testbit(i and 0x07)

proc `[]=`(bs: var BitString; i: Natural; value: bool) =
## Set the bit at index "i" to the given value.
when compileOption("boundchecks"):
checkIndex(i, bs.len)
if value: bs.values[i shr 3].setBit(i and 0x07)
else: bs.values[i shr 3].clearBit(i and 0x07)

proc fill(sieve: var BitString) =
## Fill a sieve.
var n = 0
for a in 0..1:
for b in 0..9:
for c in 0..9:
for d in 0..9:
for e in 0..9:
for f in 0..9:
for g in 0..9:
for h in 0..9:
for i in 0..9:
for j in 0..9:
sieve[a + b + c + d + e + f + g + h + i + j + n] = true
inc n

let t0 = getMonoTime()

var sieve = newBitString(MaxCount + 1)
sieve.fill()
echo "Sieve time: ", getMonoTime() - t0

# Find first 50.
echo "\nFirst 50 self numbers:"
var count = 0
var line = ""
for n in 0..MaxCount:
if not sieve[n]:
inc count
if count == 50: break
echo line

# Find 1st, 10th, 100th, ..., 100_000_000th.
echo "\n      Rank       Value"
var limit = 1
count = 0
for n in 0..MaxCount:
if not sieve[n]: inc count
if count == limit:
echo (\$count).align(10), (\$n).align(12)
limit *= 10
echo "Total time: ", getMonoTime() - t0
```
Output:
```Sieve time: 2 seconds, 59 milliseconds, 67 microseconds, and 152 nanoseconds

First 50 self numbers:
1 3 5 7 9 20 31 42 53 64 75 86 97 108 110 121 132 143 154 165 176 187 198 209 211 222 233 244 255 266 277 288 299 310 312 323 334 345 356 367 378 389 400 411 413 424 435 446 457 468

Rank       Value
1           1
10          64
100         973
1000       10188
10000      102225
100000     1022675
1000000    10227221
10000000   102272662
100000000  1022727208

Total time: 7 seconds, 903 milliseconds, 752 microseconds, and 944 nanoseconds```

### Improved sieve

Translation of: Pascal

Using bit indexing is very useful here as with byte indexing, the sieve needs 10GB. On a computer with only 8 GB , as this is the case of the laptop I use to run these programs, it fails to execute (I have a very small swap and don’t want to use the swap anyway). With bit indexing, the sieve needs only 1,25GB which is more reasonable.

Of course, the program is slower but not in the same proportion that in the previous program: it is about twice slower than the Pascal version. Note that the execution time varies significantly according to the way statements are written. For instance, writing `if not sieve[n]: inc count` has proved to be more efficient than writing `inc count, ord(not sieve[n])` or `inc count, 1 - ord(sieve[n])` which is surprising as the latter forms avoid a jump. Maybe changing some other statements could give better results, but current time is already satisfying.

```import bitops, strutils, std/monotimes, times

const MaxCount = 103 * 10_000 * 10_000 + 11 * 9 + 1

# Bit string used to represent an array of booleans.
type BitString = object
len: Natural
values: seq[byte]

proc newBitString(n: Natural): BitString  =
## Return a new bit string of length "n" bits.
result.len = n
result.values.setLen((n + 7) shr 3)

template checkIndex(i, length: Natural) {.used.} =
## Check if index "i" is less than the array length.
if i >= length:
raise newException(IndexDefect, "index \$1 not in 0 .. \$2".format(i, length))

proc `[]`(bs: BitString; i: Natural): bool =
## Return the value of bit at index "i" as a boolean.
when compileOption("boundchecks"):
checkIndex(i, bs.len)
result = bs.values[i shr 3].testbit(i and 0x07)

proc `[]=`(bs: var BitString; i: Natural; value: bool) =
## Set the bit at index "i" to the given value.
when compileOption("boundchecks"):
checkIndex(i, bs.len)
if value: bs.values[i shr 3].setBit(i and 0x07)
else: bs.values[i shr 3].clearBit(i and 0x07)

proc initDigitSum9999(): array[10000, byte] {.compileTime.} =
## Return the array of the digit sums for numbers 0 to 9999.
var i = 0
for a in 0..9:
for b in 0..9:
for c in 0..9:
for d in 0..9:
result[i] = byte(a + b + c + d)
inc i

const DigitSum9999 = initDigitSum9999()

proc fill(sieve: var BitString) =
## Fill a sieve.
var n = 0
for a in 0..102:
for b in 0..9999:
var s = DigitSum9999[a].int + DigitSum9999[b].int + n
for c in 0..9999:
sieve[DigitSum9999[c].int + s] = true
inc s
inc n, 10_000

let t0 = getMonoTime()

var sieve = newBitString(MaxCount + 1)
sieve.fill()
echo "Sieve time: ", getMonoTime() - t0

# Find first 50.
echo "\nFirst 50 self numbers:"
var count = 0
var line = ""
for n in 0..MaxCount:
if not sieve[n]:
inc count
if count == 50: break
echo line

# Find 1st, 10th, 100th, ..., 1_000_000_000th.
echo "\n      Rank       Value"
var limit = 1
count = 0
for n in 0..MaxCount:
if not sieve[n]: inc count
if count == limit:
echo (\$count).align(10), (\$n).align(12)
limit *= 10
echo "Total time: ", getMonoTime() - t0
```
Output:
```Sieve time: 13 seconds, 340 milliseconds, 45 microseconds, and 528 nanoseconds

First 50 self numbers:
1 3 5 7 9 20 31 42 53 64 75 86 97 108 110 121 132 143 154 165 176 187 198 209 211 222 233 244 255 266 277 288 299 310 312 323 334 345 356 367 378 389 400 411 413 424 435 446 457 468

Rank       Value
1           1
10          64
100         973
1000       10188
10000      102225
100000     1022675
1000000    10227221
10000000   102272662
100000000  1022727208
1000000000 10227272649
Total time: 28 seconds, 135 milliseconds, 481 microseconds, and 697 nanoseconds```

## Pascal

Works with: Free Pascal

Just "sieving" with only one follower of every number

Translation of: Go

Extended to 10.23e9

```program selfnumb;
{\$IFDEF FPC}
{\$MODE Delphi}
{\$Optimization ON,ALL}
{\$IFEND}
{\$IFDEF DELPHI} {\$APPTYPE CONSOLE} {\$IFEND}
uses
sysutils;
const
MAXCOUNT =103*10000*10000+11*9+ 1;
type
tDigitSum9999 = array[0..9999] of Uint8;
tpDigitSum9999 = ^tDigitSum9999;
var
DigitSum9999 : tDigitSum9999;
sieve : array of boolean;

procedure dosieve;
var
pSieve : pBoolean;
pDigitSum :tpDigitSum9999;
n,c,b,a,s : NativeInt;
Begin
pSieve := @sieve[0];
pDigitSum := @DigitSum9999[0];
n := 0;
for a := 0 to 102 do
for b := 0 to 9999 do
Begin
s := pDigitSum^[a]+pDigitSum^[b]+n;
for c := 0 to 9999 do
Begin
pSieve[pDigitSum^[c]+s] := true;
s+=1;
end;
inc(n,10000);
end;
end;

procedure InitDigitSum;
var
i,d,c,b,a : NativeInt;
begin
i := 9999;
for a := 9 downto 0 do
for b := 9 downto 0 do
for c := 9 downto 0 do
for d := 9 downto 0 do
Begin
DigitSum9999[i] := a+b+c+d;
dec(i);
end;
end;

procedure OutPut(cnt,i:NativeUint);
Begin
writeln(cnt:10,i:12);
end;

var
pSieve : pboolean;
T0 : Uint64;
i,cnt,limit,One: NativeUInt;
BEGIN
setlength(sieve,MAXCOUNT);
pSieve := @sieve[0];
T0 := GetTickCount64;
InitDigitSum;
dosieve;
writeln('Sievetime : ',(GetTickCount64-T0 )/1000:8:3,' sec');
//find first 50
cnt := 0;
for i := 0 to MAXCOUNT do
Begin
if NOT(pSieve[i]) then
Begin
inc(cnt);
if cnt <= 50 then
write(i:4)
else
BREAK;
end;
end;
writeln;
One := 1;
limit := One;
cnt := 0;
for i := 0 to MAXCOUNT do
Begin
inc(cnt,One-Ord(pSieve[i]));
if cnt = limit then
Begin
OutPut(cnt,i);
limit := limit*10;
end;
end;
END.
```
Output:
``` time ./selfnumb
Sievetime :    6.579 sec
1   3   5   7   9  20  31  42  53  64  75  86  97 108 110 121 132 143 154 165 176 187 198 209 211 222 233 244 255 266 277 288 299 310 312 323 334 345 356 367 378 389 400 411 413 424 435 446 457 468
1           1
10          64
100         973
1000       10188
10000      102225
100000     1022675
1000000    10227221
10000000   102272662
100000000  1022727208
1000000000 10227272649

real  0m13,252s```

## Perl

Translation of: Raku
```use strict;
use warnings;
use feature 'say';
use List::Util qw(max sum);

my (\$i, \$pow, \$digits, \$offset, \$lastSelf, @selfs)
= ( 1,   10,       1,       9,         0,       );

my \$final = 50;

while () {
my \$isSelf = 1;
my \$sum = my \$start = sum split //, max((\$i-\$offset), 0);
for ( my \$j = \$start; \$j < \$i; \$j++ ) {
if (\$j+\$sum == \$i) { \$isSelf = 0 ; last }
(\$j+1)%10 != 0 ? \$sum++ : ( \$sum = sum split '', (\$j+1) );
}

if (\$isSelf) {
push @selfs, \$lastSelf = \$i;
last if @selfs == \$final;
}

next unless ++\$i % \$pow == 0;
\$pow *= 10;
\$offset = 9 * \$digits++
}

say "The first 50 self numbers are:\n" . join ' ', @selfs;
```
Output:
```The first 50 self numbers are:
1 3 5 7 9 20 31 42 53 64 75 86 97 108 110 121 132 143 154 165 176 187 198 209 211 222 233 244 255 266 277 288 299 310 312 323 334 345 356 367 378 389 400 411 413 424 435 446 457 468```

## Phix

Translation of: AppleScript

Certainly puts my previous rubbish attempts (archived here) to shame.
The precise nature of the difference-pattern eludes me, I will admit.

```--
--  Base-10 self numbers by index (single or range).
--  Follows an observed sequence pattern whereby, after the initial single-digit odd numbers, self numbers are
--  grouped in runs whose members occur at numeric intervals of 11. Runs after the first one come in blocks of
--  ten: eight runs of ten numbers followed by two shorter runs. The numeric interval between runs is usually 2,
--  but that between shorter runs, and their length, depend on the highest-order digit change occurring in them.
--  This connection with significant digit change means every ten blocks form a higher-order block, every ten
--  of these a higher-order-still block, and so on.
--
--  The code below appears to be good up to the last self number before 10^12 — ie. 999,999,999,997, which is
--  returned as the 97,777,777,792nd such number. After this, instead of zero-length shorter runs, the actual
--  pattern apparently starts again with a single run of 10, like the one at the beginning.
--
integer startIndex, endIndex, counter
atom currentSelf
sequence output

function doneAfterAdding(integer interval, n)
-- Advance to the next self number in the sequence, append it to the output if required, indicate if finished.
for i=1 to n do
currentSelf += interval
counter += 1
if counter >= startIndex then
output &= currentSelf
end if
if counter = endIndex then return true end if
end for
return false
end function

function selfNumbers(sequence indexRange)
startIndex = indexRange[1]
endIndex = indexRange[\$]
counter = 0
currentSelf = -1
output = {}

-- Main process. Start with the single-digit odd numbers and first run.
if doneAfterAdding(2,5) then return output end if
if doneAfterAdding(11,9) then return output end if

-- If necessary, fast forward to last self number before the lowest-order block containing first number rqd.
if counter<startIndex then
-- The highest-order blocks whose ends this handles correctly contain 9,777,777,778 self numbers.
-- The difference between equivalently positioned numbers in these blocks is 100,000,000,001.
-- The figures for successively lower-order blocks have successively fewer 7s and 0s!
atom indexDiff = 9777777778,
numericDiff = 100000000001
while indexDiff>=98 and counter!=startIndex do
if counter+indexDiff < startIndex then
counter += indexDiff
currentSelf += numericDiff
else
indexDiff = (indexDiff+2)/10    -- (..78->80->8)
numericDiff = (numericDiff+9)/10 -- (..01->10->1)
end if
end while
end if

-- Sequencing loop, per lowest-order block.
while true do
-- Eight ten-number runs, each at a numeric interval of 2 from the end of the previous one.
for i=1 to 8 do
if doneAfterAdding(2,1) then return output end if
if doneAfterAdding(11,9) then return output end if
end for
-- Two shorter runs, the second at an interval inversely related to their length.
integer shorterRunLength = 8,
temp = floor(currentSelf/1000)
-- Work out a shorter run length based on the most significant digit change about to happen.
while remainder(temp,10)=9 do
shorterRunLength -= 1
temp = floor(temp/10)
end while

integer interval = 2
for i=1 to 2 do
if doneAfterAdding(interval,1) then return output end if
if doneAfterAdding(11,shorterRunLength) then return output end if
interval += (9-shorterRunLength)*13
end for
end while
end function

atom t0 = time()
printf(1,"The first 50 self numbers are:\n")
pp(selfNumbers({1, 50}),{pp_IntFmt,"%3d",pp_IntCh,false})
for p=8 to 9 do
integer n = power(10,p)
printf(1,"The %,dth safe number is %,d\n",{n,selfNumbers({n})[1]})
end for
printf(1,"completed in %s\n",elapsed(time()-t0))
```
Output:
```The first 50 self numbers are:
{  1,  3,  5,  7,  9, 20, 31, 42, 53, 64, 75, 86, 97,108,110,121,132,143,
154,165,176,187,198,209,211,222,233,244,255,266,277,288,299,310,312,323,
334,345,356,367,378,389,400,411,413,424,435,446,457,468}
The 100,000,000th safe number is 1,022,727,208
The 1,000,000,000th safe number is 10,227,272,649
completed in 0.1s
```

## Python

Works with: Python version 2.7
```class DigitSumer :
def __init__(self):
sumdigit = lambda n : sum( map( int,str( n )))
self.t = [sumdigit( i ) for i in xrange( 10000 )]
def __call__ ( self,n ):
r = 0
while n >= 10000 :
n,q = divmod( n,10000 )
r += self.t[q]
return r + self.t[n]

def self_numbers ():
d = DigitSumer()
s = set([])
i = 1
while 1 :
n = i + d( i )
if i in s :
else:
yield i
i += 1

import time
p = 100
t = time.time()
for i,s in enumerate( self_numbers(),1 ):
if i <= 50 :
print s,
if i == 50 : print
if i == p :
print '%7.1f sec  %9dth = %d'%( time.time()-t,i,s )
p *= 10
```
Output:
```1 3 5 7 9 20 31 42 53 64 75 86 97 108 110 121 132 143 154 165 176 187 198 209 211 222 233 244 255 266 277 288 299 310 312 323 334 345 356 367 378 389 400 411 413 424 435 446 457 468
0.0 sec        100th = 973
0.0 sec       1000th = 10188
0.1 sec      10000th = 102225
1.0 sec     100000th = 1022675
11.4 sec    1000000th = 10227221
143.4 sec   10000000th = 102272662
1812.0 sec  100000000th = 1022727208```

## Raku

Translated the low memory version of the Go entry but showed only the first 50 self numbers. The machine for running this task (a Xeon E3110+8GB memory) is showing its age as, 1) took over 6 minutes to complete the Go entry, 2) not even able to run the other two Go alternative entries and 3) needed over 47 minutes to reach 1e6 iterations here. Anyway I will try this on an i5 box later to see how it goes.

Translation of: Go
```# 20201127 Raku programming solution

my ( \$st, \$count, \$i, \$pow, \$digits, \$offset, \$lastSelf, \$done, @selfs) =
now,      0,  1,   10,       1,       9,         0, False;

# while ( \$count < 1e8 ) {
until \$done {
my \$isSelf = True;
my \$sum    = (my \$start = max (\$i-\$offset), 0).comb.sum;
loop ( my \$j = \$start; \$j < \$i; \$j+=1 ) {
if \$j+\$sum == \$i { \$isSelf = False and last }
(\$j+1)%10 != 0 ?? ( \$sum+=1 ) !! ( \$sum = (\$j+1).comb.sum ) ;
}
if \$isSelf {
\$count+=1;
\$lastSelf = \$i;
if \$count ≤ 50 {
@selfs.append: \$i;
if \$count == 50 {
say "The first 50 self numbers are:\n", @selfs;
\$done = True;
}
}
}
\$i+=1;
if \$i % \$pow == 0 {
\$pow *= 10;
\$digits+=1 ;
\$offset = \$digits * 9
}
}

# say "The 100 millionth self number is ", \$lastSelf;
# say "Took ", now - \$st, " seconds."
```
Output:
```The first 50 self numbers are:
[1 3 5 7 9 20 31 42 53 64 75 86 97 108 110 121 132 143 154 165 176 187 198 209 211 222 233 244 255 266 277 288 299 310 312 323 334 345 356 367 378 389 400 411 413 424 435 446 457 468]```

## REXX

### first 50 self numbers

```/*REXX program displays  N  self numbers (aka Colombian or Devlali numbers). OEIS A3052.*/
parse arg n .                                    /*obtain optional argument from the CL.*/
if n=='' | n==","  then n= 50                    /*Not specified?  Then use the default.*/
tell = n>0;             n= abs(n)                /*TELL:  show the self numbers if  N>0 */
@.= .                                            /*initialize the array of self numbers.*/
do j=1  for n*10                      /*scan through ten times the #s wanted.*/
\$= j                                  /*1st part of sum is the number itself.*/
do k=1  for length(j)           /*sum the decimal digits in the number.*/
\$= \$ + substr(j, k, 1)          /*add a particular digit to the sum.   */
end   /*k*/
@.\$=                                  /*mark  J  as not being a self number. */
end         /*j*/                     /*            ───                      */
list= 1                                          /*initialize the list to the 1st number*/
#= 1                            /*the count of self numbers  (so far). */
do i=3  until #==n;  if @.i=='' then iterate  /*Not a self number?   Then skip it.   */
#= # + 1;            list= list i             /*bump counter of self #'s; add to list*/
end   /*i*/
/*stick a fork in it,  we're all done. */
say   n     " self numbers were found."          /*display the title for the output list*/
if tell  then say list                           /*display list of self numbers ──►term.*/
```
output   when using the default input:
```50  self numbers were found.
1 3 5 7 9 20 31 42 53 64 75 86 97 108 110 121 132 143 154 165 176 187 198 209 211 222 233 244 255 266 277 288 299 310 312 323 334 345 356 367 378 389 400 411 413 424 435 446 457 468
```

### ten millionth self number

Translation of: Go (low memory)
```/*REXX pgm displays the  Nth  self number, aka Colombian or Devlali numbers. OEIS A3052.*/
numeric digits 20                                /*ensure enough decimal digits for #.  */
parse arg high .                                 /*obtain optional argument from the CL.*/
if high=='' | high==","  then high= 10000000     /*Not specified?  Then use 10M default.*/
i= 1;   pow= 10;   digs= 1;    offset= 9;   \$= 0 /*\$:  the last self number found.      */
#= 0                                             /*count of self numbers  (so far).     */
do while #<high;          isSelf= 1         /*assume a self number   (so far).     */
start= max(i-offset, 0)                     /*find start #;  must be non-negative. */
sum= sumDigs(start)                         /*obtain the sum of the decimal digits.*/

do j=start  to i-1
if j+sum==i  then do;  isSelf= 0         /*found a   non  self number.          */
iterate           /*keep looking for more self numbers.  */
end
if (j+1)//10==0   then sum= sumDigs(j+1) /*obtain the sum of the decimal digits.*/
else sum= sum + 1      /*bump    "   "   "  "     "      "    */
end   /*j*/

if isSelf  then do;  #= # + 1               /*bump the count of self numbers.      */
\$= i                   /*save the last self number found.     */
end
i= i + 1                                    /*bump the self number by unity.       */
if i//pow==0  then do;    digs= digs + 1    /*  "   "  number of decimal digits.   */
pow= pow * 10    /*  "   "  power   " a factor of ten.  */
offset= digs * 9    /*  "   "  offset  " "    "    " nine. */
end
end   /*while*/
say
say 'the '   commas(high)th(high)     " self number is: "     commas(\$)
exit 0                                           /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
sumDigs: parse arg s 2 x;   do k=1  for length(x)   /*get 1st dig,  & also get the rest.*/
s= s + substr(x, k, 1)  /*add a particular digit to the sum.*/
end  /*k*/;  return s
/*──────────────────────────────────────────────────────────────────────────────────────*/
commas:  parse arg _;  do c=length(_)-3  to 1  by -3; _=insert(',', _, c); end;   return _
th:      parse arg th; return word('th st nd rd', 1 +(th//10)*(th//100%10\==1)*(th//10<4))
```
output   when using the default input:
```the  100,000,000th  self number is:  1,022,727,208
```

## Ring

```load "stdlib.ring"

see "working..." + nl
see "The first 50 self numbers are:" + nl

n = 0
num = 0
limit = 51
limit2 = 10000000

while true
n = n + 1
for m = 1 to n
flag = 1
sum = 0
strnum = string(m)
for p = 1 to len(strnum)
sum = sum + number(strnum[p])
next
sum2 = m + sum
if sum2 = n
flag = 0
exit
else
flag = 1
ok
next
if flag = 1
num = num + 1
if num < limit
see "" + num + ". " + n + nl
ok
if num = limit2
see "The " + limit2 + "th self number is: " + n + nl
ok
if num > limit2
exit
ok
ok
end

see "done..." + nl```

Output:

```working...
The first 50 self numbers are:
1. 1
2. 3
3. 5
4. 7
5. 9
6. 20
7. 31
8. 42
9. 53
10. 64
11. 75
12. 86
13. 97
14. 108
15. 110
16. 121
17. 132
18. 143
19. 154
20. 165
21. 176
22. 187
23. 198
24. 209
25. 211
26. 222
27. 233
28. 244
29. 255
30. 266
31. 277
32. 288
33. 299
34. 310
35. 312
36. 323
37. 334
38. 345
39. 356
40. 367
41. 378
42. 389
43. 400
44. 411
45. 413
46. 424
47. 435
48. 446
49. 457
50. 468
The 10000000th self number is: 1022727208
done...
```

## RPL

#### Brute force

Using a sieve:

```« 0
WHILE OVER REPEAT
SWAP 10 MOD LASTARG / IP
ROT ROT +
END +
» 'DIGSUM' STO

« 500 0 → max n
« { } DUP max + 0 CON
1 CF
DO 'n' INCR
DUP DIGSUM +
IFERR 1 PUT THEN DROP2 1 SF END
UNTIL 1 FS? END
1
WHILE 3 PICK SIZE 50 < REPEAT
IF DUP2 GET NOT THEN ROT OVER + ROT ROT END
1 +
END DROP2
» » 'TASK' STO
```
Output:
```1: { 1 3 5 7 9 20 31 42 53 64 75 86 97 108 110 121 132 143 154 165 176 187 198 209 211 222 233 244 255 266 277 288 299 310 312 323 334 345 356 367 378 389 400 411 413 424 435 446 457 468 }
```

Runs in 2 minutes 8 seconds on a HP-48SX.

#### Maximilian F. Hasler's algorithm

Translation of the PARI/GP formula on the OEIS page:

```« → n
« { } 1 SF
1 n 2 / IP n XPON 1 + 9 * MIN FOR j
IF n j - DIGSUM j == THEN 1 CF n 'j' STO END
NEXT
1 FS?
» 'SELF?' STO

« { }
WHILE OVER SIZE 50 < REPEAT
IF DUP SELF? THEN SWAP OVER + SWAP END
1 +
END DROP
```

Same result as above. No need for sieve, but much slower: 10 minutes 52 seconds on a HP-48SX.

## Sidef

Algorithm by David A. Corneth (OEIS: A003052).

```func is_self_number(n) {

if (n < 30) {
return (((n < 10) && (n.is_odd)) || (n == 20))
}

var qd = (1 + n.ilog10)
var r  = (1 + (n-1)%9)
var h  = (r + 9*(r%2))/2
var ld = 10

while (h + 9*qd >= n%ld) {
ld *= 10
}

var vs = idiv(n, ld).sumdigits
n %= ld

0..qd -> none { |i|
vs + sumdigits(n - h - 9*i) == (h + 9*i)
}
}

say is_self_number.first(50).join(' ')
```
Output:
```1 3 5 7 9 20 31 42 53 64 75 86 97 108 110 121 132 143 154 165 176 187 198 209 211 222 233 244 255 266 277 288 299 310 312 323 334 345 356 367 378 389 400 411 413 424 435 446 457 468
```

Simpler algorithm (by M. F. Hasler):

```func is_self_number(n) {
1..min(n>>1, 9*n.len) -> none {|i| sumdigits(n-i) == i } && (n > 0)
}
```

## Standard ML

```open List;

val rec selfNumberNr = fn NR =>
let
val rec sumdgt = fn 0 => 0 | n => Int.rem (n, 10) + sumdgt (Int.quot(n ,10));
val rec isSelf  = fn ([],l1,l2) => []
| (x::tt,l1,l2) => if exists (fn i=>i=x) l1 orelse exists (fn i=>i=x) l2
then ( isSelf (tt,l1,l2)) else x::isSelf (tt,l1,l2) ;

val rec partcount =  fn  (n, listIn , count , selfs) =>
if count >= NR then  nth (selfs, length selfs + NR - count -1)
else
let
val range   = tabulate (81 , fn i => 81*n +i+1) ;
val listOut = map (fn i => i + sumdgt i ) range ;
val selfs   = isSelf (range,listIn,listOut)
in
partcount ( n+1 , listOut , count+length (selfs) , selfs )
end;
in
partcount (0,[],0,[])
end;

app  ((fn s => print (s ^ " ")) o Int.toString o selfNumberNr)  (tabulate (50,fn i=>i+1)) ;
selfNumberNr 100000000 ;
```

output

```1 3 5 7 9 20 31 42 53 64 75 86 97 108 110 121 132 143 154 165 176 187 198 209 211 222 233 244 255 266 277 288 299 310 312 323 334 345 356 367 378 389 400 411 413 424 435 446 457 468
1022727208
```

## Wren

Translation of: Go

Just the sieve based version as the low memory version would take too long to run in Wren.

Note that you need a lot of memory to run this as Bools in Wren require 8 bytes of storage compared to 1 byte in Go.

Unsurprisingly, very slow compared to the Go version as Wren is interpreted and uses floating point arithmetic for all numerical work.

```var sieve = Fn.new {
var sv = List.filled(2*1e9+9*9+1, false)
var n = 0
var s = [0] * 8
for (a in 0..1) {
for (b in 0..9) {
s[0] = a + b
for (c in 0..9) {
s[1] = s[0] + c
for (d in 0..9) {
s[2] = s[1] + d
for (e in 0..9) {
s[3] = s[2] + e
for (f in 0..9) {
s[4] = s[3] + f
for (g in 0..9) {
s[5] = s[4] + g
for (h in 0..9) {
s[6] = s[5] + h
for (i in 0..9) {
s[7] = s[6] + i
for (j in 0..9) {
sv[s[7] + j + n] = true
n = n + 1
}
}
}
}
}
}
}
}
}
}
return sv
}

var st = System.clock
var sv = sieve.call()
var count = 0
System.print("The first 50 self numbers are:")
for (i in 0...sv.count) {
if (!sv[i]) {
count = count + 1
if (count <= 50) System.write("%(i) ")
if (count == 1e8) {
System.print("\n\nThe 100 millionth self number is %(i)")
break
}
}
}
System.print("Took %(System.clock-st) seconds.")
```
Output:
```The first 50 self numbers are:
1 3 5 7 9 20 31 42 53 64 75 86 97 108 110 121 132 143 154 165 176 187 198 209 211 222 233 244 255 266 277 288 299 310 312 323 334 345 356 367 378 389 400 411 413 424 435 446 457 468

The 100 millionth self number is 1022727208
Took 222.789713 seconds.
```

## XPL0

Translation of: Go

As run on Raspberry Pi4.

```def  LenSV = 2_000_000_000 + 9*9 + 1;

func Sieve;
char SV;
int  N, S(8), A, B, C, D, E, F, G, H, I, J;
[SV:= MAlloc(LenSV);
N:= 0;
for A:= 0 to 1 do
[for B:= 0 to 9 do
[S(0):= A + B;
for C:= 0 to 9 do
[S(1):= S(0) + C;
for D:= 0 to 9 do
[S(2):= S(1) + D;
for E:= 0 to 9 do
[S(3):= S(2) + E;
for F:= 0 to 9 do
[S(4):= S(3) + F;
for G:= 0 to 9 do
[S(5):= S(4) + G;
for H:= 0 to 9 do
[S(6):= S(5) + H;
for I:= 0 to 9 do
[S(7):= S(6) + I;
for J:= 0 to 9 do
[SV(S(7)+J+N):= true;
N:= N+1;
]
]
]
]
]
]
]
]
]
];
return SV;
];

char SV;
int  ST, Count, I;
[ST:= GetTime;
SV:= Sieve;
Count:= 0;
Text(0, "The first 50 self numbers are:^m^j");
for I:= 0 to LenSV-1 do
[if SV(I) = false then
[Count:= Count+1;
if Count <= 50 then
[IntOut(0, I);  ChOut(0, ^ )];
if Count = 100_000_000 then
[Text(0, "^m^j^m^jThe 100 millionth self number is ");
IntOut(0, I);
CrLf(0);
I:= LenSV;
];
];
];
Text(0, "Took ");  RlOut(0, float(GetTime-ST) / 1e6);  CrLf(0);
]```
Output:
```The first 50 self numbers are:
1 3 5 7 9 20 31 42 53 64 75 86 97 108 110 121 132 143 154 165 176 187 198 209 211 222 233 244 255 266 277 288 299 310 312 323 334 345 356 367 378 389 400 411 413 424 435 446 457 468

The 100 millionth self number is 1022727208
Took    29.46575
```