# Ludic numbers

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

Ludic numbers are related to prime numbers as they are generated by a sieve quite like the Sieve of Eratosthenes is used to generate prime numbers.

The first ludic number is 1.
To generate succeeding ludic numbers create an array of increasing integers starting from 2

`2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 ...`

(Loop)

• Take the first member of the resultant array as the next Ludic number 2.
• Remove every 2'nd indexed item from the array (including the first).
`2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 ...`
• (Unrolling a few loops...)
• Take the first member of the resultant array as the next Ludic number 3.
• Remove every 3'rd indexed item from the array (including the first).
`3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 ...`
• Take the first member of the resultant array as the next Ludic number 5.
• Remove every 5'th indexed item from the array (including the first).
`5 7 11 13 17 19 23 25 29 31 35 37 41 43 47 49 53 55 59 61 65 67 71 73 77 ...`
• Take the first member of the resultant array as the next Ludic number 7.
• Remove every 7'th indexed item from the array (including the first).
`7 11 13 17 23 25 29 31 37 41 43 47 53 55 59 61 67 71 73 77 83 85 89 91 97 ...`
• ...
• Take the first member of the current array as the next Ludic number L.
• Remove every L'th indexed item from the array (including the first).
• ...
• Generate and show here the first 25 ludic numbers.
• How many ludic numbers are there less than or equal to 1000?
• Show the 2000..2005'th ludic numbers.
• A triplet is any three numbers ${\displaystyle x,}$ ${\displaystyle x+2,}$ ${\displaystyle x+6}$ where all three numbers are also ludic numbers. Show all triplets of ludic numbers < 250 (Stretch goal)

## ABAP

Works with NW 7.40 SP8

`CLASS lcl_ludic DEFINITION CREATE PUBLIC.   PUBLIC SECTION.    TYPES: t_ludics TYPE SORTED TABLE OF i WITH UNIQUE KEY table_line.    TYPES: BEGIN OF t_triplet,             i1 TYPE i,             i2 TYPE i,             i3 TYPE i,           END OF t_triplet.    TYPES: t_triplets TYPE STANDARD TABLE OF t_triplet WITH EMPTY KEY.     CLASS-METHODS:      ludic_up_to        IMPORTING i_int           TYPE i        RETURNING VALUE(r_ludics) TYPE t_ludics,      get_triplets        IMPORTING i_ludics          TYPE t_ludics        RETURNING VALUE(r_triplets) TYPE t_triplets.     "RETURNING parameters (CallByValue) only used for readability of the demo    "in "Real Life" you should use EXPORTING (CallByRef) for tables ENDCLASS. cl_demo_output=>begin_section( 'First 25 Ludics' ).cl_demo_output=>write( lcl_ludic=>ludic_up_to( 110 ) ). cl_demo_output=>begin_section( 'Ludics up to 1000' ).cl_demo_output=>write( lines( lcl_ludic=>ludic_up_to( 1000 ) ) ). cl_demo_output=>begin_section( '2000th - 2005th Ludics' ).DATA(ludics) = lcl_ludic=>ludic_up_to( 22000 ).cl_demo_output=>write( VALUE lcl_ludic=>t_ludics( FOR i = 2000 WHILE i <= 2005 ( ludics[ i ] ) ) ). cl_demo_output=>begin_section( 'Triplets up to 250' ).cl_demo_output=>write( lcl_ludic=>get_triplets( lcl_ludic=>ludic_up_to( 250 ) ) ). cl_demo_output=>display( ). CLASS lcl_ludic IMPLEMENTATION.   METHOD ludic_up_to.     r_ludics = VALUE #( FOR i = 2 WHILE i <= i_int ( i ) ).     DATA(cursor) = 0.     WHILE cursor < lines( r_ludics ).       cursor = cursor + 1.      DATA(this_ludic) = r_ludics[ cursor ].      DATA(remove_cursor) = cursor + this_ludic.       WHILE remove_cursor <= lines( r_ludics ).        DELETE r_ludics INDEX remove_cursor.        remove_cursor = remove_cursor + this_ludic - 1.      ENDWHILE.     ENDWHILE.     INSERT 1 INTO TABLE r_ludics.  "add one as the first Ludic number (per definition)   ENDMETHOD.   METHOD get_triplets.     DATA(i) = 0.    WHILE i < lines( i_ludics ) - 2.      i = i + 1.       DATA(this_ludic) = i_ludics[ i ].      IF  line_exists( i_ludics[ table_line = this_ludic + 2 ] )      AND line_exists( i_ludics[ table_line = this_ludic + 6 ] ).        r_triplets = VALUE #(           BASE r_triplets           ( i1 = i_ludics[ table_line = this_ludic ]             i2 = i_ludics[ table_line = this_ludic + 2 ]             i3 = i_ludics[ table_line = this_ludic + 6 ]           )        ).      ENDIF.     ENDWHILE.   ENDMETHOD. ENDCLASS.`
Output:
```First 25 Ludics
1
2
3
5
7
11
13
17
23
25
29
37
41
43
47
53
61
67
71
77
83
89
91
97
107

Ludics up to 1000
142

2000th - 2005th Ludics
21475
21481
21487
21493
21503
21511

Triplets up to 250
1 3 7
5 7 11
11 13 17
23 25 29
41 43 47
173 175 179
221 223 227
233 235 239
```

## AutoHotkey

Works with: AutoHotkey 1.1
`#NoEnvSetBatchLines, -1Ludic := LudicSieve(22000) Loop, 25    ; the first 25 ludic numbers	Task1 .= Ludic[A_Index] " " for i, Val in Ludic    ; the number of ludic numbers less than or equal to 1000	if (Val <= 1000)		Task2++	else		break Loop, 6    ; the 2000..2005'th ludic numbers	Task3 .= Ludic[1999 + A_Index] " " for i, Val in Ludic {    ; all triplets of ludic numbers < 250	if (Val + 6 > 249)		break	if (Ludic[i + 1] = Val + 2 && Ludic[i + 2] = Val + 6 || i = 1)		Task4 .= "(" Val " " Val + 2 " " Val + 6 ") "} MsgBox, % "First 25:`t`t" Task1	. "`nLudics below 1000:`t" Task2	. "`nLudic 2000 to 2005:`t" Task3	. "`nTriples below 250:`t" Task4return LudicSieve(Limit) {	Arr := [], Ludic := []	Loop, % Limit		Arr.Insert(A_Index)	Ludic.Insert(Arr.Remove(1))	while Arr.MaxIndex() != 1 {		Ludic.Insert(n := Arr.Remove(1))		, Removed := 0		Loop, % Arr.MaxIndex() // n {			Arr.Remove(A_Index * n - Removed)			, Removed++		}	}	Ludic.Insert(Arr[1])	return Ludic}`
Output:
```First 25:		1 2 3 5 7 11 13 17 23 25 29 37 41 43 47 53 61 67 71 77 83 89 91 97 107
Ludics below 1000:	142
Ludic 2000 to 2005:	21475 21481 21487 21493 21503 21511
Triples below 250:	(1 3 7) (5 7 11) (11 13 17) (23 25 29) (41 43 47) (173 175 179) (221 223 227) (233 235 239) ```

## C

`#include <stdio.h>#include <stdlib.h> typedef unsigned uint;typedef struct { uint i, v; } filt_t; // ludics with at least so many elements and reach at least such valueuint* ludic(uint min_len, uint min_val, uint *len){	uint cap, i, v, active = 1, nf = 0;	filt_t *f = calloc(cap = 2, sizeof(*f));	f[1].i = 4; 	for (v = 1; ; ++v) {		for (i = 1; i < active && --f[i].i; i++); 		if (i < active)			f[i].i = f[i].v;		else if (nf == f[i].i)			f[i].i = f[i].v, ++active;  // enable one more filter		else {			if (nf >= cap)				f = realloc(f, sizeof(*f) * (cap*=2));			f[nf] = (filt_t){ v + nf, v };			if (++nf >= min_len && v >= min_val) break;		}	} 	// pack the sequence into a uint[]	// filt_t struct was used earlier for cache locality in loops	uint *x = (void*) f;	for (i = 0; i < nf; i++) x[i] = f[i].v;	x = realloc(x, sizeof(*x) * nf); 	*len = nf;	return x;} int find(uint *a, uint v){	uint i;	for (i = 0; a[i] <= v; i++)		if (v == a[i]) return 1;	return 0;} int main(void){	uint len, i, *x = ludic(2005, 1000, &len); 	printf("First 25:");	for (i = 0; i < 25; i++) printf(" %u", x[i]);	putchar('\n'); 	for (i = 0; x[i] <= 1000; i++);	printf("Ludics below 1000: %u\n", i); 	printf("Ludic 2000 to 2005:");	for (i = 2000; i <= 2005; i++) printf(" %u", x[i - 1]);	putchar('\n'); 	printf("Triples below 250:");	for (i = 0; x[i] + 6 <= 250; i++)		if (find(x, x[i] + 2) && find(x, x[i] + 6))			printf(" (%u %u %u)", x[i], x[i] + 2, x[i] + 6); 	putchar('\n'); 	free(x);	return 0;}`
Output:
```First 25: 1 2 3 5 7 11 13 17 23 25 29 37 41 43 47 53 61 67 71 77 83 89 91 97 107
Ludics below 1000: 142
Ludic 2000 to 2005: 21475 21481 21487 21493 21503 21511
Triples below 250: (1 3 7) (5 7 11) (11 13 17) (23 25 29) (41 43 47) (173 175 179) (221 223 227) (233 235 239)
```

## C++

` #include <vector>#include <iostream>using namespace std; class ludic{public:    void ludicList()    {        _list.push_back( 1 );         vector<int> v;        for( int x = 2; x < 22000; x++ )            v.push_back( x );         while( true )        {            vector<int>::iterator i = v.begin();            int z = *i;            _list.push_back( z );             while( true )            {                i = v.erase( i );                if( distance( i, v.end() ) <= z - 1 ) break;                advance( i, z - 1 );            }            if( v.size() < 1 ) return;        }    }     void show( int s, int e )    {        for( int x = s; x < e; x++ )            cout << _list[x] << " ";    }     void findTriplets( int e )    {        int lu, x = 0;        while( _list[x] < e )        {            lu = _list[x];            if( inList( lu + 2 ) && inList( lu + 6 ) )                cout << "(" << lu << " " << lu + 2 << " " << lu + 6 << ")\n";            x++;        }    }     int count( int e )    {        int x = 0, c = 0;        while( _list[x++] <= 1000 ) c++;        return c;    } private:    bool inList( int lu )    {        for( int x = 0; x < 250; x++ )            if( _list[x] == lu ) return true;        return false;    }     vector<int> _list;}; int main( int argc, char* argv[] ){    ludic l;    l.ludicList();    cout << "first 25 ludic numbers:" << "\n";    l.show( 0, 25 );    cout << "\n\nThere are " << l.count( 1000 ) << " ludic numbers <= 1000" << "\n";    cout << "\n2000 to 2005'th ludic numbers:" << "\n";    l.show( 1999, 2005 );    cout << "\n\nall triplets of ludic numbers < 250:" << "\n";    l.findTriplets( 250 );    cout << "\n\n";    return system( "pause" );} `
Output:
```first 25 ludic numbers:
1 2 3 5 7 11 13 17 23 25 29 37 41 43 47 53 61 67 71 77 83 89 91 97 107

There are 142 ludic numbers <= 1000

2000 to 2005'th ludic numbers:
21475 21481 21487 21493 21503 21511

all triplets of ludic numbers < 250:
(1 3 7)
(5 7 11)
(11 13 17)
(23 25 29)
(41 43 47)
(173 175 179)
(221 223 227)
(233 235 239)
```

## Clojure

`(defn ints-from [n]  (cons n (lazy-seq (ints-from (inc n))))) (defn drop-nth [n seq]    (cond       (zero?    n) seq      (empty? seq) []      :else (concat (take (dec n) seq) (lazy-seq (drop-nth n (drop n seq)))))) (def ludic ((fn ludic   ([] (ludic 1))   ([n] (ludic n (ints-from (inc n))))   ([n [f & r]] (cons n (lazy-seq (ludic f (drop-nth f r)))))))) (defn ludic? [n]  (= (first (filter (partial <= n) ludic)) n)) (print "First 25: ")(println (take 25 ludic))(print "Count below 1000: ")(println (count (take-while (partial > 1000) ludic)))(print "2000th through 2005th: ")(println (map (partial nth ludic) (range 1999 2005)))(print "Triplets < 250: ")(println (filter (partial every? ludic?)          (for [i (range 250)] (list i (+ i 2) (+ i 6)))))`
Output:
```First 25: (1 2 3 5 7 11 13 17 23 25 29 37 41 43 47 53 61 67 71 77 83 89 91 97 107)
Count below 1000: 142
2000th through 2005th: (21475 21481 21487 21493 21503 21511)
Triplets < 250: ((1 3 7) (5 7 11) (11 13 17) (23 25 29) (41 43 47) (173 175 179) (221 223 227) (233 235 239))```

## D

### opApply Version

Translation of: Python
Translation of: Perl 6
`struct Ludics(T) {    int opApply(int delegate(in ref T) dg) {        int result;        T[] rotor, taken = [T(1)];        result = dg(taken[0]);        if (result) return result;         for (T i = 2; ; i++) { // Shoud be stopped if T has a max.            size_t j = 0;            for (; j < rotor.length; j++)                if (!--rotor[j])                    break;             if (j < rotor.length) {                rotor[j] = taken[j + 1];            } else {                result = dg(i);                if (result) return result;                taken ~= i;                rotor ~= taken[j + 1];            }        }    }} void main() {    import std.stdio, std.range, std.algorithm;     // std.algorithm.take can't be used here.    uint[] L;    foreach (const x; Ludics!uint())        if (L.length < 2005)            L ~= x;        else            break;     writeln("First 25 ludic primes:\n", L.take(25));    writefln("\nThere are %d ludic numbers <= 1000.",             L.until!q{ a > 1000 }.walkLength);     writeln("\n2000'th .. 2005'th ludic primes:\n", L[1999 .. 2005]);     enum m = 250;    const triplets = L.filter!(x => x + 6 < m &&                                    L.canFind(x + 2) && L.canFind(x + 6))                     // Ugly output:                     //.map!(x => tuple(x, x + 2, x + 6)).array;                     .map!(x => [x, x + 2, x + 6]).array;    writefln("\nThere are %d triplets less than %d:\n%s",             triplets.length, m, triplets);}`
Output:
```First 25 ludic primes:
[1, 2, 3, 5, 7, 11, 13, 17, 23, 25, 29, 37, 41, 43, 47, 53, 61, 67, 71, 77, 83, 89, 91, 97, 107]

There are 142 ludic numbers <= 1000.

2000'th .. 2005'th ludic primes:
[21475, 21481, 21487, 21493, 21503, 21511]

There are 8 triplets less than 250:
[[1, 3, 7], [5, 7, 11], [11, 13, 17], [23, 25, 29], [41, 43, 47], [173, 175, 179], [221, 223, 227], [233, 235, 239]]```

The run-time is about 0.03 seconds or less. It takes about 2.0 seconds to generate 50_000 Ludic numbers with ldc2 compiler.

### Range Version

This is the same code modified to be a Range.

`struct Ludics(T) {    T[] rotor, taken = [T(1)];    T i;    size_t j;    T front = 1; // = taken[0];    bool running = false;    static immutable bool empty = false;     void popFront() pure nothrow @safe {        if (running)            goto RESUME;        else            running = true;         i = 2;        while (true) {            j = 0;             while (j < rotor.length) {                rotor[j]--;                if (!rotor[j])                    break;                j++;            }            if (j < rotor.length) {                rotor[j] = taken[j + 1];            } else {                front = i;                return;        RESUME:                taken ~= i;                rotor ~= taken[j + 1];            }            i++; // Could overflow if T has a max.        }    }} void main() {    import std.stdio, std.range, std.algorithm, std.array;     Ludics!uint L;    writeln("First 25 ludic primes:\n", L.take(25));    writefln("\nThere are %d ludic numbers <= 1000.",             L.until!q{ a > 1000 }.walkLength);     writeln("\n2000'th .. 2005'th ludic primes:\n", L.drop(1999).take(6));     enum uint m = 250;    const few = L.until!(x => x > m).array;    const triplets = few.filter!(x => x + 6 < m && few.canFind(x + 2)                                      && few.canFind(x + 6))                     // Ugly output:                     //.map!(x => tuple(x, x + 2, x + 6)).array;                     .map!(x => [x, x + 2, x + 6]).array;    writefln("\nThere are %d triplets less than %d:\n%s",             triplets.length, m, triplets);}`

The output is the same. This version is slower, it takes about 3.3 seconds to generate 50_000 Ludic numbers with ldc2 compiler.

### Range Generator Version

`void main() {    import std.stdio, std.range, std.algorithm, std.concurrency;     Generator!T ludics(T)() {        return new typeof(return)({            T[] rotor, taken = [T(1)];            yield(taken[0]);             for (T i = 2; ; i++) { // Shoud be stopped if T has a max.                size_t j = 0;                for (; j < rotor.length; j++)                    if (!--rotor[j])                        break;                 if (j < rotor.length) {                    rotor[j] = taken[j + 1];                } else {                    yield(i);                    taken ~= i;                    rotor ~= taken[j + 1];                }            }        });    }     const L = ludics!uint.take(2005).array;     writeln("First 25 ludic primes:\n", L.take(25));    writefln("\nThere are %d ludic numbers <= 1000.",             L.until!q{ a > 1000 }.walkLength);     writeln("\n2000'th .. 2005'th ludic primes:\n", L[1999 .. 2005]);     enum m = 250;    const triplets = L.filter!(x => x + 6 < m &&                                    L.canFind(x + 2) && L.canFind(x + 6))                     // Ugly output:                     //.map!(x => tuple(x, x + 2, x + 6)).array;                     .map!(x => [x, x + 2, x + 6]).array;    writefln("\nThere are %d triplets less than %d:\n%s",             triplets.length, m, triplets);}`

The result is the same.

## Eiffel

` class	LUDIC_NUMBERS create	make feature 	make (n: INTEGER)			-- Initialized arrays for find_ludic_numbers.		require			n_positive: n > 0		local			i: INTEGER		do			create initial.make_filled (0, 1, n - 1)			create ludic_numbers.make_filled (1, 1, 1)			from				i := 2			until				i > n			loop				initial.put (i, i - 1)				i := i + 1			end			find_ludic_numbers		end 	ludic_numbers: ARRAY [INTEGER] feature {NONE} 	initial: ARRAY [INTEGER] 	find_ludic_numbers			-- Ludic numbers in array ludic_numbers.		local			count: INTEGER			new_array: ARRAY [INTEGER]			last: INTEGER		do			create new_array.make_from_array (initial)			last := initial.count			from				count := 1			until				count > last			loop				if ludic_numbers [ludic_numbers.count] /= new_array [1] then					ludic_numbers.force (new_array [1], count + 1)				end				new_array := delete_i_elements (new_array)				count := count + 1			end		end 	delete_i_elements (ar: ARRAY [INTEGER]): ARRAY [INTEGER]			--- Array with all multiples of 'ar[1]' deleted.		require			ar_not_empty: ar.count > 0		local			s_array: ARRAY [INTEGER]			i, k: INTEGER			length: INTEGER		do			create s_array.make_empty			length := ar.count			from				i := 0				k := 1			until				i = length			loop				if (i) \\ (ar [1]) /= 0 then					s_array.force (ar [i + 1], k)					k := k + 1				end				i := i + 1			end			if s_array.count = 0 then				Result := ar			else				Result := s_array			end		ensure			not_empty: not Result.is_empty		end end `

Test:

` class	APPLICATION create	make feature 	make		local			k, count: INTEGER		do			create ludic.make (22000)			io.put_string ("%NLudic numbers up to 25. %N")			across				ludic.ludic_numbers.subarray (1, 25) as ld			loop				io.put_string (ld.item.out + "%N")			end			io.put_string ("%NLudic numbers from 2000 ... 2005. %N")			across				ludic.ludic_numbers.subarray (2000, 2005) as ld			loop				io.put_string (ld.item.out + "%N")			end			io.put_string ("%NNumber of Ludic numbers smaller than 1000. %N")			from				k := 1			until				ludic.ludic_numbers [k] >= 1000			loop				k := k + 1				count := count + 1			end			io.put_integer (count)		end 	ludic: LUDIC_NUMBERS end `
Output:
```Ludic numbers up to 25.
1 2 3 5 7 11 13 17 23 25 29 37 41 43 47 53 61 67 71 77 83 89 91 97 107

Ludic numbers from 2000 ... 2005.
21475
21481
21487
21493
21503
21511

Number of Ludic numbers smaller than 1000.
142
```

## Elixir

`defmodule Ludic do  def numbers, do: numbers(100000)   def numbers(n) when is_integer(n) do    [h|t] = Enum.to_list(1..n)    numbers(t, [h])  end   defp numbers(list, nums) when length(list) < hd(list), do: Enum.reverse(nums, list)  defp numbers(list, nums) do    h = hd(list)    ludic = Enum.with_index(list) |>             Enum.filter_map(fn{_,i} -> rem(i,h)!=0 end, fn{n,_} -> n end)    numbers(ludic, [h | nums])  end   def task do    IO.puts "First 25 : #{inspect numbers(200) |> Enum.take(25)}"    IO.puts "Below 1000: #{length(numbers(1000))}"    tuple = numbers(25000) |> List.to_tuple    IO.puts "2000..2005th: #{ inspect Enum.map(1999..2004, fn i -> elem(tuple, i) end) }"    ludic = numbers(250)    triple = for x<-ludic, Enum.member?(ludic, x+2), Enum.member?(ludic, x+6), do: [x, x+2, x+6]    IO.puts "Triples below 250: #{inspect triple, char_lists: :as_lists}"  endend Ludic.task`
Output:
```First 25 : [1, 2, 3, 5, 7, 11, 13, 17, 23, 25, 29, 37, 41, 43, 47, 53, 61, 67, 71, 77, 83, 89, 91, 97, 107]
Below 1000: 142
2000..2005th: [21475, 21481, 21487, 21493, 21503, 21511]
Triples below 250: [[1, 3, 7], [5, 7, 11], [11, 13, 17], [23, 25, 29], [41, 43, 47], [173, 175, 179], [221, 223, 227], [233, 235, 239]]
```

## Fortran

Works with: Fortran version 95 and later
`program ludic_numbers  implicit none   integer, parameter :: nmax = 25000  logical :: ludic(nmax) = .true.  integer :: i, j, n   do i = 2, nmax / 2    if (ludic(i)) then      n = 0      do j = i+1, nmax        if(ludic(j)) n = n + 1        if(n == i) then          ludic(j) = .false.          n = 0        end if      end do    end if  end do   write(*, "(a)", advance = "no") "First 25 Ludic numbers: "  n = 0  do i = 1, nmax    if(ludic(i)) then      write(*, "(i0, 1x)", advance = "no") i      n = n + 1    end if    if(n == 25) exit  end do     write(*, "(/, a)", advance = "no") "Ludic numbers below 1000: "  write(*, "(i0)") count(ludic(:999))   write(*, "(a)", advance = "no") "Ludic numbers 2000 to 2005: "   n = 0  do i = 1, nmax    if(ludic(i)) then       n = n + 1       if(n >= 2000) then         write(*, "(i0, 1x)", advance = "no") i         if(n == 2005) exit       end if     end if  end do     write(*, "(/, a)", advance = "no") "Ludic Triplets below 250: "  do i = 1, 243    if(ludic(i) .and. ludic(i+2) .and. ludic(i+6)) then       write(*, "(a, 2(i0, 1x), i0, a, 1x)", advance = "no") "[", i, i+2, i+6, "]"    end if    end do end program`

Output:

```First 25 Ludic numbers: 1 2 3 5 7 11 13 17 23 25 29 37 41 43 47 53 61 67 71 77 83 89 91 97 107
Ludic numbers below 1000: 142
Ludic numbers 2000 to 2005: 21475 21481 21487 21493 21503 21511
Ludic Triplets below 250: [1 3 7] [5 7 11] [11 13 17] [23 25 29] [41 43 47] [173 175 179] [221 223 227] [233 235 239]```

## Go

`package main import "fmt" // Ludic returns a slice of Ludic numbers stopping after// either n entries or when max is exceeded.// Either argument may be <=0 to disable that limit.func Ludic(n int, max int) []uint32 {	const maxInt32 = 1<<31 - 1 // i.e. math.MaxInt32	if max > 0 && n < 0 {		n = maxInt32	}	if n < 1 {		return nil	}	if max < 0 {		max = maxInt32	}	sieve := make([]uint32, 10760) // XXX big enough for 2005 Ludics	sieve[0] = 1	sieve[1] = 2	if n > 2 {		// We start with even numbers already removed		for i, j := 2, uint32(3); i < len(sieve); i, j = i+1, j+2 {			sieve[i] = j		}		// We leave the Ludic numbers in place,		// k is the index of the next Ludic		for k := 2; k < n; k++ {			l := int(sieve[k])			if l >= max {				n = k				break			}			i := l			l--			// last is the last valid index			last := k + i - 1			for j := k + i + 1; j < len(sieve); i, j = i+1, j+1 {				last = k + i				sieve[last] = sieve[j]				if i%l == 0 {					j++				}			}			// Truncate down to only the valid entries			if last < len(sieve)-1 {				sieve = sieve[:last+1]			}		}	}	if n > len(sieve) {		panic("program error") // should never happen	}	return sieve[:n]} func has(x []uint32, v uint32) bool {	for i := 0; i < len(x) && x[i] <= v; i++ {		if x[i] == v {			return true		}	}	return false} func main() {	// Ludic() is so quick we just call it repeatedly	fmt.Println("First 25:", Ludic(25, -1))	fmt.Println("Numner of Ludics below 1000:", len(Ludic(-1, 1000)))	fmt.Println("Ludic 2000 to 2005:", Ludic(2005, -1)[1999:]) 	fmt.Print("Tripples below 250:")	x := Ludic(-1, 250)	for i, v := range x[:len(x)-2] {		if has(x[i+1:], v+2) && has(x[i+2:], v+6) {			fmt.Printf(", (%d %d %d)", v, v+2, v+6)		}	}	fmt.Println()}`
Output:
```First 25: [1 2 3 5 7 11 13 17 23 25 29 37 41 43 47 53 61 67 71 77 83 89 91 97 107]
Numner of Ludics below 1000: 142
Ludic 2000 to 2005: [21475 21481 21487 21493 21503 21511]
Tripples below 250:, (1 3 7), (5 7 11), (11 13 17), (23 25 29), (41 43 47), (173 175 179), (221 223 227), (233 235 239)```

`import Data.List (unfoldr, genericSplitAt) ludic :: [Integer]ludic = 1 : unfoldr (\xs@(x:_) -> Just (x, dropEvery x xs)) [2..] where  dropEvery n = concat . map tail . unfoldr (Just . genericSplitAt n) main :: IO ()main = do  print \$ take 25 \$ ludic  print \$ length \$ takeWhile (<= 1000) \$ ludic  print \$ take 6 \$ drop 1999 \$ ludic  -- haven't done triplets task yet`
Output:
```[1,2,3,5,7,11,13,17,23,25,29,37,41,43,47,53,61,67,71,77,83,89,91,97,107]
142
[21475,21481,21487,21493,21503,21511]
```

The filter for dropping every n-th number can be delayed until it's needed, which speeds up the generator, more so when a longer sequence is taken.

`ludic = 1:2 : f 3 [3..] [(4,2)] where	f n (x:xs) yy@((i,y):ys)		| n == i = f n (dropEvery y xs) ys		| otherwise = x : f (1+n) xs (yy ++ [(n+x, x)]) dropEvery n s = a ++ dropEvery n (tail b) where	(a,b) = splitAt (n-1) s main = print \$ ludic !! 10000`

## Icon and Unicon

This is inefficient, but was fun to code as a cascade of filters. Works in both languages.

`global num, cascade, sieve, nfilter procedure main(A)    lds := ludic(2005)		# All we need for the four tasks.    every writes("First 25:" | (" "||!lds)\25 | "\n")    every (n := 0) +:= (!lds < 1000, 1)    write("There are ",n," Ludic numbers < 1000.")    every writes("2000th through 2005th: " | (lds[2000 to 20005]||" ") | "\n")    writes("Triplets:")    every (250 > (x := !lds)) & (250 > (x+2 = !lds)) & (250 > (x+6 = !lds)) do        writes(" [",x,",",x+2,",",x+6,"]")    write()end procedure ludic(limit)    candidates := create seq(2)    put(cascade := [], create {        repeat {            report(l := num, limit)            put(cascade, create (cnt:=0, repeat ((cnt+:=1)%l=0, @sieve) | @@nfilter))            cascade[-2] :=: cascade[-1]  # keep this sink as the last filter            @sieve            }        })    sieve := create while num := @candidates do @@(nfilter := create !cascade)    report(1, limit)    return @sieveend procedure report(ludic, limit)    static count, lds    initial {count := 0; lds := []}    if (count +:= 1) > limit then lds@&main    put(lds, ludic)end`

Output:

```->ludic
First 25: 1 2 3 5 7 11 13 17 23 25 29 37 41 43 47 53 61 67 71 77 83 89 91 97 107
There are 142 Ludic numbers < 1000.
2000th through 20005th: 21475 21481 21487 21493 21503 21511
Triplets: [1,3,7] [5,7,11] [11,13,17] [23,25,29] [41,43,47] [173,175,179] [221,223,227] [233,235,239]
->
```

## J

Solution (naive / brute force):
`   ludic =: _1 |.!.1 [: {."1 [: (#~ 0 ~: {. | i.@#)^:a: 2 + i.`
Examples:
`   # ludic 110  NB. 110 is sufficient to generate 25 Ludic numbers25   ludic 110    NB. First 25 Ludic numbers1 2 3 5 7 11 13 17 23 25 29 37 41 43 47 53 61 67 71 77 83 89 91 97 107    #ludic 1000  NB. 142 Ludic numbers <= 1000142    # ludic 22000   NB. 22000 is sufficient to generate > 2005 Ludic numbers2042   (2000+i.6) { ludic 22000  NB. Ludic numbers 2000-200521481 21487 21493 21503 21511 21523    0 2 6 (] (*./ .e.~ # |:@]) +/) ludic 250  NB. Ludic triplets <= 250  1   3   7  5   7  11 11  13  17 23  25  29 41  43  47173 175 179221 223 227233 235 239`

## Java

Works with: Java version 1.5+

This example uses pre-calculated ranges for the first and third task items (noted in comments).

`import java.util.ArrayList;import java.util.List; public class Ludic{	public static List<Integer> ludicUpTo(int n){		List<Integer> ludics = new ArrayList<Integer>(n);		for(int i = 1; i <= n; i++){   //fill the initial list			ludics.add(i);		} 		//start at index 1 because the first ludic number is 1 and we don't remove anything for it		for(int cursor = 1; cursor < ludics.size(); cursor++){			int thisLudic = ludics.get(cursor); //the first item in the list is a ludic number			int removeCursor = cursor + thisLudic; //start removing that many items later			while(removeCursor < ludics.size()){				ludics.remove(removeCursor);		     //remove the next item				removeCursor = removeCursor + thisLudic - 1; //move the removal cursor up as many spaces as we need to									     //then back one to make up for the item we just removed			}		}		return ludics;	} 	public static List<List<Integer>> getTriplets(List<Integer> ludics){		List<List<Integer>> triplets = new ArrayList<List<Integer>>();		for(int i = 0; i < ludics.size() - 2; i++){ //only need to check up to the third to last item			int thisLudic = ludics.get(i);			if(ludics.contains(thisLudic + 2) && ludics.contains(thisLudic + 6)){				List<Integer> triplet = new ArrayList<Integer>(3);				triplet.add(thisLudic);				triplet.add(thisLudic + 2);				triplet.add(thisLudic + 6);				triplets.add(triplet);			}		}		return triplets;	} 	public static void main(String[] srgs){		System.out.println("First 25 Ludics: " + ludicUpTo(110));				//110 will get us 25 numbers		System.out.println("Ludics up to 1000: " + ludicUpTo(1000).size());		System.out.println("2000th - 2005th Ludics: " + ludicUpTo(22000).subList(1999, 2005));  //22000 will get us 2005 numbers		System.out.println("Triplets up to 250: " + getTriplets(ludicUpTo(250)));	}}`
Output:
```First 25 Ludics: [1, 2, 3, 5, 7, 11, 13, 17, 23, 25, 29, 37, 41, 43, 47, 53, 61, 67, 71, 77, 83, 89, 91, 97, 107]
Ludics up to 1000: 142
2000th - 2005th Ludics: [21475, 21481, 21487, 21493, 21503, 21511]
Triplets up to 250: [[1, 3, 7], [5, 7, 11], [11, 13, 17], [23, 25, 29], [41, 43, 47], [173, 175, 179], [221, 223, 227], [233, 235, 239]]```

## Julia

` function ludic_filter{T<:Integer}(n::T)    0 < n || throw(DomainError())    slud = trues(n)    for i in 2:(n-1)        slud[i] || continue        x = 0        for j in (i+1):n            slud[j] || continue            x += 1            x %= i            x == 0 || continue            slud[j] = false        end    end    return sludend ludlen = 10^5slud = ludic_filter(ludlen)ludics = collect(1:ludlen)[slud] n = 25println("Generate and show here the first ", n, " ludic numbers.")print("    ")crwid = 76wid = 0for i in 1:(n-1)    s = @sprintf "%d, " ludics[i]    wid += length(s)    if crwid < wid        print("\n    ")        wid = 0    end    print(s)endprintln(ludics[n]) n = 10^3println()println("How many ludic numbers are there less than or equal to ", n, "?")println("    ", sum(slud[1:n])) lo = 2000hi = lo+5println()println("Show the ", lo, "..", hi, "'th ludic numbers.")for i in lo:hi    println("    Ludic(", i, ") = ", ludics[i])end n = 250println()println("Show all triplets of ludic numbers < ", n)for i = 1:n-7    slud[i] || continue    j = i+2    slud[j] || continue    k = i+6    slud[k] || continue    println("    ", i, ", ", j, ", ", k)end `
Output:
```Generate and show here the first 25 ludic numbers.
1, 2, 3, 5, 7, 11, 13, 17, 23, 25, 29, 37, 41, 43, 47, 53, 61, 67, 71, 77,
83, 89, 91, 97, 107

How many ludic numbers are there less than or equal to 1000?
142

Show the 2000..2005'th ludic numbers.
Ludic(2000) = 21475
Ludic(2001) = 21481
Ludic(2002) = 21487
Ludic(2003) = 21493
Ludic(2004) = 21503
Ludic(2005) = 21511

Show all triplets of ludic numbers < 250
1, 3, 7
5, 7, 11
11, 13, 17
23, 25, 29
41, 43, 47
173, 175, 179
221, 223, 227
233, 235, 239
```

## Mathematica

`n=10^5;Ludic={1};seq=Range[2,n];ClearAll[DoStep]DoStep[seq:{f_,___}]:=Module[{out=seq}, AppendTo[Ludic,f]; out[[;;;;f]]=Sequence[]; out]Nest[DoStep,seq,2500];`
Output:
```Ludic[[;; 25]]
LengthWhile[Ludic, # < 1000 &]
Ludic[[2000 ;; 2005]]
Select[Subsets[Select[Ludic, # < 250 &], {3}], Differences[#] == {2, 4} &]

{1, 2, 3, 5, 7, 11, 13, 17, 23, 25, 29, 37, 41, 43, 47, 53, 61, 67, 71, 77, 83, 89, 91, 97, 107}
142
{21475, 21481, 21487, 21493, 21503, 21511}
{{1, 3, 7}, {5, 7, 11}, {11, 13, 17}, {23, 25, 29}, {41, 43, 47}, {173, 175, 179}, {221, 223, 227}, {233, 235, 239}}```

## Oforth

`func: ludic(n){| ludics l p |   ListBuffer newSize(n) seqFrom(2, n) over addAll ->l   ListBuffer newSize(n) dup add(1) ->ludics    while(l notEmpty) [      l removeFirst dup ludics add ->p        l size p / p * while(dup 1 > ) [ dup l removeAt drop p - ] drop      ]    ludics} func: ludics{| l i |   ludic(22000) ->l   "First 25     : " print l left(25) println   "Below 1000   : " print l filter(#[ 1000 < ]) size println   "2000 to 2005 : " print l extract(2000, 2005) println    250 loop: i [      l include(i) ifFalse: [ continue ]      l include(i 2 +) ifFalse: [ continue ]      l include(i 6 +) ifFalse: [ continue ]      i print ", " print i 2 + print ", " print i 6 + println      ]}`
Output:
```First 25     : [1, 2, 3, 5, 7, 11, 13, 17, 23, 25, 29, 37, 41, 43, 47, 53, 61, 67, 71, 77, 83, 89, 91, 97, 107]
Below 1000   : 142
2000 to 2005 : [21475, 21481, 21487, 21493, 21503, 21511]
1, 3, 7
5, 7, 11
11, 13, 17
23, 25, 29
41, 43, 47
173, 175, 179
221, 223, 227
233, 235, 239
```

## PARI/GP

Works with: PARI/GP version 2.7.4 and above
` \\ Creating Vlf - Vector of ludic numbers' flags,\\ where the index of each flag=1 is the ludic number.ludic(maxn)={my(Vlf=vector(maxn,z,1),n2=maxn/2,k,j1);for(i=2,n2,     if(Vlf[i], k=0; j1=i+1;       for(j=j1,maxn, if(Vlf[j], k++); if(k==i, Vlf[j]=0; k=0))      );    );return(Vlf);} {\\ Required tests:my(Vr,L=List(),k=0,maxn=25000);Vr=ludic(maxn);print("The first 25 Ludic numbers: ");for(i=1,maxn, if(Vr[i]==1, k++; print1(i," "); if(k==25, break)));print("");print("");k=0;for(i=1,999, if(Vr[i]==1, k++));print("Ludic numbers below 1000: ",k);print("");k=0;print("Ludic numbers 2000 to 2005: ");for(i=1,maxn, if(Vr[i]==1, k++; if(k>=2000&&k<=2005, listput(L,i)); if(k>2005, break)));for(i=1,6, print1(L[i]," "));print(""); print("");print("Ludic Triplets below 250: ");for(i=1,250, if(Vr[i]&&Vr[i+2]&&Vr[i+6], print1("(",i," ",i+2," ",i+6,") ")));} `
Output:
```The first 25 Ludic numbers:
1 2 3 5 7 11 13 17 23 25 29 37 41 43 47 53 61 67 71 77 83 89 91 97 107

Ludic numbers below 1000: 142

Ludic numbers 2000 to 2005:
21475 21481 21487 21493 21503 21511

Ludic Triplets below 250:
(1 3 7) (5 7 11) (11 13 17) (23 25 29) (41 43 47) (173 175 179) (221 223 227) (233 235 239)
```

## Pascal

Inspired by "rotors" of perl 6 . Runtime nearly quadratic: maxLudicCnt = 10000 -> 0.03 s =>maxLudicCnt= 100000 -> 3 s

`program lucid;{\$IFDEF FPC}  {\$MODE objFPC} // useful for x64{\$ENDIF} const  //66164 -> last < 1000*1000;  maxLudicCnt = 2005;//must be > 1type   tDelta = record             dNum,             dCnt : LongInt;           end;   tpDelta = ^tDelta;  tLudicList = array of tDelta;   tArrdelta =array[0..0] of tDelta;  tpLl = ^tArrdelta; function isLudic(plL:tpLl;maxIdx:nativeInt):boolean;var  i,  cn : NativeInt;Begin  //check if n is 'hit' by a prior ludic number  For i := 1 to maxIdx do    with plL^[i] do    Begin      //Mask read modify write reread      //dec(dCnt);IF dCnt= 0      cn := dCnt;      IF cn = 1 then      Begin        dcnt := dNum;        isLudic := false;        EXIT;       end;      dcnt := cn-1;    end;  isLudic := true;end; procedure CreateLudicList(var Ll:tLudicList);var  plL : tpLl;  n,LudicCnt : NativeUint;begin  // special case 1  n := 1;  Ll[0].dNum := 1;   plL := @Ll[0];  LudicCnt := 0;  repeat    inc(n);    If isLudic(plL,LudicCnt ) then    Begin      inc(LudicCnt);      with plL^[LudicCnt] do      Begin        dNum := n;        dCnt := n;      end;      IF (LudicCnt >= High(LL)) then        BREAK;    end;  until false;end; procedure  firstN(var Ll:tLudicList;cnt: NativeUint);var  i : NativeInt;Begin  writeln('First ',cnt,' ludic numbers:');  For i := 0 to cnt-2 do    write(Ll[i].dNum,',');  writeln(Ll[cnt-1].dNum);end; procedure triples(var Ll:tLudicList;max: NativeUint);var  i,  chk : NativeUint;Begin  // special case 1,3,7  writeln('Ludic triples below ',max);  write('(',ll[0].dNum,',',ll[2].dNum,',',ll[4].dNum,') ');   For i := 1 to High(Ll) do  Begin    chk := ll[i].dNum;    If chk> max then      break;    If (ll[i+2].dNum = chk+6) AND (ll[i+1].dNum = chk+2) then      write('(',ll[i].dNum,',',ll[i+1].dNum,',',ll[i+2].dNum,') ');  end;  writeln;  writeln;end; procedure LastLucid(var Ll:tLudicList;start,cnt: NativeUint);var  limit,i : NativeUint;Begin  dec(start);  limit := high(Ll);  IF cnt >= limit then    cnt := limit;  if start+cnt >limit then    start := limit-cnt;  writeln(Start+1,'.th to ',Start+cnt+1,'.th ludic number');  For i := 0 to cnt-1 do    write(Ll[i+start].dNum,',');  writeln(Ll[start+cnt].dNum);  writeln;end; function CountLudic(var Ll:tLudicList;Limit: NativeUint):NativeUint;var  i,res : NativeUint;Begin  res := 0;  For i := 0 to High(Ll) do begin    IF Ll[i].dnum <= Limit then      inc(res)    else      BREAK;  CountLudic:= res;end; end;var  LudicList : tLudicList;BEGIN  setlength(LudicList,maxLudicCnt);  CreateLudicList(LudicList);  firstN(LudicList,25);  writeln('There are ',CountLudic(LudicList,1000),' ludic numbers below 1000');  LastLucid(LudicList,2000,5);  LastLucid(LudicList,maxLudicCnt,5);  triples(LudicList,250);//all-> (LudicList,LudicList[High(LudicList)].dNum);END.{1,2,3,5,7,11,13,17,23,25,29,37,41,43,47,53,61,67,71,77,83,89,91,97,107There are 142 ludic numbers below 10002000.th to 2005.th ludic number21475,21481,21487,21493,21503,21511 99995.th to 100000.th ludic number1561243,1561291,1561301,1561307,1561313,1561333 Ludic triples below 250(1,3,7) (5,7,11) (11,13,17) (23,25,29) (41,43,47) (173,175,179) (221,223,227) (233,235,239) real  0m2.921s}`
Output:
```First 25 ludic numbers:
1,2,3,5,7,11,13,17,23,25,29,37,41,43,47,53,61,67,71,77,83,89,91,97,107
There are 142 ludic numbers below 1000
2000.th to 2005.th ludic number
21481,21487,21493,21503,21511

Ludic triples below 250
(1,3,7) (5,7,11) (11,13,17) (23,25,29) (41,43,47) (173,175,179) (221,223,227) (233,235,239)
```

## Perl

The "ludic" subroutine caches the longest generated sequence so far. It also generates the candidates only if no candidates remain.

`#!/usr/bin/perluse warnings;use strict;use feature qw{ say }; {   my @ludic = (1);    my \$max = 3;    my @candidates;     sub sieve {        my \$l = shift;        for (my \$i = 0; \$i <= \$#candidates; \$i += \$l) {            splice @candidates, \$i, 1;        }    }     sub ludic {        my (\$type, \$n) = @_;        die "Arg0 Type must be 'count' or 'max'\n"             unless grep \$_ eq \$type, qw( count max );        die "Arg1 Number must be > 0\n" if 0 >= \$n;         return (@ludic[ 0 .. \$n - 1 ]) if 'count' eq \$type and @ludic >= \$n;         return (grep \$_ <= \$n, @ludic) if 'max'   eq \$type and \$ludic[-1] >= \$n;         while (1) {            if (@candidates) {                last if ('max' eq \$type and \$candidates[0] > \$n)                     or (\$n == @ludic);                 push @ludic, \$candidates[0];                sieve(\$ludic[-1] - 1);             } else {                \$max *= 2;                @candidates = 2 .. \$max;                for my \$l (@ludic) {                    sieve(\$l - 1) unless 1 == \$l;                }            }        }        return (@ludic)    } } my @triplet;my %ludic;undef @ludic{ ludic(max => 250) };for my \$i (keys %ludic) {    push @triplet, \$i if exists \$ludic{ \$i + 2 } and exists \$ludic { \$i + 6 };} say 'First 25:       ', join ' ', ludic(count => 25);say 'Count < 1000:   ', scalar ludic(max => 1000);say '2000..2005th:   ', join ' ', (ludic(count => 2005))[1999 .. 2004];say 'triplets < 250: ', join ' ',                        map { '(' . join(' ',\$_, \$_ + 2, \$_ + 6) . ')' }                        sort { \$a <=> \$b } @triplet;`
Output:
```First 25:       1 2 3 5 7 11 13 17 23 25 29 37 41 43 47 53 61 67 71 77 83 89 91 97 107
Count < 1000:   142
2000..2005th:   21475 21481 21487 21493 21503 21511
triplets < 250: (1 3 7) (5 7 11) (11 13 17) (23 25 29) (41 43 47) (173 175 179) (221 223 227) (233 235 239)```

## Perl 6

Works with: rakudo version 2015-09-18

This implementation has no arbitrary upper limit, since it can keep adding new rotors on the fly. It just gets slower and slower instead... :-)

`constant @ludic = gather {        my @taken = take 1;        my @rotor;         for 2..* -> \$i {            loop (my \$j = 0; \$j < @rotor; \$j++) {                --@rotor[\$j] or last;            }            if \$j < @rotor {                @rotor[\$j] = @taken[\$j+1];            }            else {                push @taken, take \$i;                push @rotor, @taken[\$j+1];            }        }    } say @ludic[^25];say "Number of Ludic numbers <= 1000: ", +(@ludic ...^ * > 1000);say "Ludic numbers 2000..2005: ", @ludic[1999..2004]; my \l250 = set @ludic ...^ * > 250;say "Ludic triples < 250: ", gather    for l250.keys.sort -> \$a {        my \$b = \$a + 2;        my \$c = \$a + 6;        take "<\$a \$b \$c>" if \$b ∈ l250 and \$c ∈ l250;    }`
Output:
```(1 2 3 5 7 11 13 17 23 25 29 37 41 43 47 53 61 67 71 77 83 89 91 97 107)
Number of Ludic numbers <= 1000: 142
Ludic numbers 2000..2005: (21475 21481 21487 21493 21503 21511)
Ludic triples < 250: (<1 3 7> <5 7 11> <11 13 17> <23 25 29> <41 43 47> <173 175 179> <221 223 227> <233 235 239>)```

## PicoLisp

`(de drop (Lst)   (let N (car Lst)      (make         (for (I . X) (cdr Lst)            (unless (=0 (% I N)) (link X)) ) ) ) ) (de comb (M Lst)   (cond      ((=0 M) '(NIL))      ((not Lst))      (T         (conc            (mapcar               '((Y) (cons (car Lst) Y))               (comb (dec M) (cdr Lst)) )            (comb M (cdr Lst)) ) ) ) ) (de ludic (N)   (let Ludic (range 1 100000)      (make         (link (pop 'Ludic))         (do (dec N)            (link (car Ludic))            (setq Ludic (drop Ludic)) ) ) ) ) (let L (ludic 2005)   (println (head 25 L))   (println (cnt '((X) (< X 1000)) L))   (println (tail 6 L))   (println      (filter         '((Lst)            (and              (= (+ 2 (car Lst)) (cadr Lst))              (= (+ 6 (car Lst)) (caddr Lst)) ) )         (comb            3            (filter '((X) (< X 250)) L) ) ) ) ) (bye)`
Output:
```
(1 2 3 5 7 11 13 17 23 25 29 37 41 43 47 53 61 67 71 77 83 89 91 97 107)
142
(21475 21481 21487 21493 21503 21511)

((1 3 7) (5 7 11) (11 13 17) (23 25 29) (41 43 47) (173 175 179) (221 223 227) (233 235 239))```

## PL/I

 This example is incorrect. Missing Triplet 1,3,7. Please fix the code and remove this message.
`Ludic_numbers: procedure options (main);                /* 18 April 2014 */   declare V(2:22000) fixed, L(2200) fixed;   declare (step, i, j, k, n) fixed binary; Ludic: procedure;   n = hbound(V,1); k = 1; L(1) = 1;   do i = 2 to n; V(i) = i; end;    do forever;       k = k + 1; L(k), step = V(2);       do i = 2 to n by step;         V(i) = 0;      end;      call compress;      if L(k) >= 21511 then leave;   end;    put skip list ('The first 25 Ludic numbers are:');   put skip edit ( (L(i) do i = 1 to 25) ) (F(4));    k = 0;   do i = 1 by 1;      if L(i) < 1000 then k = k + 1; else leave;   end;    put skip list ('There are ' || trim(k) || ' Ludic numbers < 1000');   put skip list ('Six Ludic numbers from the 2000-th:');   put skip edit ( (L(i) do i = 2000 to 2005) ) (f(7));   /* Triples are values of the form x, x+2, x+6 */   put skip list ('Triples are:');   put skip;   i = 1;   do i = 1 by 1 while (L(i+2) <= 250);      if (L(i) = L(i+1) - 2) & (L(i) = L(i+2) - 6) then         put edit ('(', L(i), L(i+1), L(i+2), ') ' ) (A, 3 F(4), A);   end; compress: procedure;   j = 2;   do i = 2 to n;      if V(i) ^= 0 then do; V(j) = V(i); j = j + 1; end;   end;   n = j-1;end compress; end Ludic; call Ludic; end Ludic_numbers;`

Output:

```The first 25 Ludic numbers are:
1   2   3   5   7  11  13  17  23  25  29  37  41  43  47
53  61  67  71  77  83  89  91  97 107
There are 142 Ludic numbers < 1000
Six Ludic numbers from the 2000-th:
21475  21481  21487  21493  21503  21511
Triples are:
(   5   7  11) (  11  13  17) (  23  25  29) (  41  43  47)
( 173 175 179) ( 221 223 227) ( 233 235 239)```

## PL/SQL

`SET SERVEROUTPUT ONDECLARE  c_limit CONSTANT PLS_INTEGER := 25000;  TYPE t_nums IS TABLE OF PLS_INTEGER INDEX BY PLS_INTEGER;  v_nums t_nums;  v_ludic t_nums;  v_count_ludic PLS_INTEGER;  v_count_pos PLS_INTEGER;  v_pos PLS_INTEGER;  v_next_ludic PLS_INTEGER;   FUNCTION is_ludic(p_num PLS_INTEGER) RETURN BOOLEAN IS  BEGIN    FOR i IN 1..v_ludic.COUNT LOOP      EXIT WHEN v_ludic(i) > p_num;      IF v_ludic(i) = p_num THEN        RETURN TRUE;      END IF;    END LOOP;    RETURN FALSE;  END; BEGIN  FOR i IN 1..c_limit LOOP    v_nums(i) := i;  END LOOP;   v_count_ludic := 1;  v_next_ludic := 1;  v_ludic(v_count_ludic) := v_next_ludic;  v_nums.DELETE(1);   WHILE v_nums.COUNT > 0 LOOP    v_pos := v_nums.FIRST;    v_next_ludic := v_nums(v_pos);    v_count_ludic := v_count_ludic + 1;    v_ludic(v_count_ludic) := v_next_ludic;    v_count_pos := 0;    WHILE v_pos IS NOT NULL LOOP      IF MOD(v_count_pos, v_next_ludic) = 0 THEN        v_nums.DELETE(v_pos);      END IF;      v_pos := v_nums.NEXT(v_pos);      v_count_pos := v_count_pos + 1;    END LOOP;  END LOOP;   DBMS_OUTPUT.put_line('Generate and show here the first 25 ludic numbers.');  FOR i IN 1..25 LOOP    DBMS_OUTPUT.put(v_ludic(i) || ' ');  END LOOP;  DBMS_OUTPUT.put_line('');   DBMS_OUTPUT.put_line('How many ludic numbers are there less than or equal to 1000?');  v_count_ludic := 0;  FOR i IN 1..v_ludic.COUNT LOOP    EXIT WHEN v_ludic(i) > 1000;    v_count_ludic := v_count_ludic + 1;  END LOOP;  DBMS_OUTPUT.put_line(v_count_ludic);   DBMS_OUTPUT.put_line('Show the 2000..2005''th ludic numbers.');  FOR i IN 2000..2005 LOOP    DBMS_OUTPUT.put(v_ludic(i) || ' ');  END LOOP;  DBMS_OUTPUT.put_line('');   DBMS_OUTPUT.put_line('A triplet is any three numbers x, x + 2, x + 6 where all three numbers are also ludic numbers.');  DBMS_OUTPUT.put_line('Show all triplets of ludic numbers < 250 (Stretch goal)');  FOR i IN 1..v_ludic.COUNT LOOP    EXIT WHEN (v_ludic(i)+6) >= 250;    IF is_ludic(v_ludic(i)+2) AND is_ludic(v_ludic(i)+6) THEN      DBMS_OUTPUT.put_line(v_ludic(i) || ', ' || (v_ludic(i)+2) || ', ' || (v_ludic(i)+6));    END IF;  END LOOP; END;/ `
Output:
```Generate and show here the first 25 ludic numbers.
1 2 3 5 7 11 13 17 23 25 29 37 41 43 47 53 61 67 71 77 83 89 91 97 107
How many ludic numbers are there less than or equal to 1000?
142
Show the 2000..2005'th ludic numbers.
21475 21481 21487 21493 21503 21511
A triplet is any three numbers x, x + 2, x + 6 where all three numbers are also ludic numbers.
Show all triplets of ludic numbers < 250 (Stretch goal)
1, 3, 7
5, 7, 11
11, 13, 17
23, 25, 29
41, 43, 47
173, 175, 179
221, 223, 227
233, 235, 239```

## Python

### Python: Fast

`def ludic(nmax=100000):    yield 1    lst = list(range(2, nmax + 1))    while lst:        yield lst[0]        del lst[::lst[0]] ludics = [l for l in ludic()] print('First 25 ludic primes:')print(ludics[:25])print("\nThere are %i ludic numbers <= 1000"      % sum(1 for l in ludics if l <= 1000)) print("\n2000'th..2005'th ludic primes:")print(ludics[2000-1: 2005]) n = 250triplets = [(x, x+2, x+6)            for x in ludics            if x+6 < n and x+2 in ludics and x+6 in ludics]print('\nThere are %i triplets less than %i:\n  %r'      % (len(triplets), n, triplets))`
Output:
```First 25 ludic primes:
[1, 2, 3, 5, 7, 11, 13, 17, 23, 25, 29, 37, 41, 43, 47, 53, 61, 67, 71, 77, 83, 89, 91, 97, 107]

There are 142 ludic numbers <= 1000

2000'th..2005'th ludic primes:
[21475, 21481, 21487, 21493, 21503, 21511]

There are 8 triplets less than 250:
[(1, 3, 7), (5, 7, 11), (11, 13, 17), (23, 25, 29), (41, 43, 47), (173, 175, 179), (221, 223, 227), (233, 235, 239)]```

### Python: No set maximum

The following version of function ludic will return ludic numbers until reaching system limits. It is less efficient than the fast version as all lucid numbers so far are cached; on exhausting the current lst a new list of twice the size is created and the previous deletions applied before continuing.

`def ludic(nmax=64):    yield 1    taken = []    while True:        lst, nmax = list(range(2, nmax + 1)), nmax * 2        for t in taken:            del lst[::t]        while lst:            t = lst[0]            taken.append(t)            yield t            del lst[::t]`

Output is the same as for the fast version.

## Racket

`#lang racket(define lucid-sieve-size 25000) ; this should be enough to do me!(define lucid?  (let ((lucid-bytes-sieve         (delay           (define sieve-bytes (make-bytes lucid-sieve-size 1))                  (bytes-set! sieve-bytes 0 0) ; not a lucid number           (define (sieve-pass L)             (let loop ((idx (add1 L)) (skip (sub1 L)))               (cond                 [(= idx lucid-sieve-size)                  (for/first ((rv (in-range (add1 L) lucid-sieve-size))                              #:unless (zero? (bytes-ref sieve-bytes rv))) rv)]                 [(zero? (bytes-ref sieve-bytes idx))                  (loop (add1 idx) skip)]                 [(= skip 0)                  (bytes-set! sieve-bytes idx 0)                  (loop (add1 idx) (sub1 L))]                 [else (loop (add1 idx) (sub1 skip))])))           (let loop ((l 2))             (when l (loop (sieve-pass l))))           sieve-bytes)))     (λ (n) (= 1 (bytes-ref (force lucid-bytes-sieve) n))))) (define (dnl . things) (for-each displayln things)) (dnl "Generate and show here the first 25 ludic numbers." (for/list ((_ 25) (l (sequence-filter lucid? (in-naturals)))) l) "How many ludic numbers are there less than or equal to 1000?" (for/sum ((n 1001) #:when (lucid? n)) 1) "Show the 2000..2005'th ludic numbers." (for/list ((i 2006) (l (sequence-filter lucid? (in-naturals))) #:when (>= i 2000)) l) #<<EOSA triplet is any three numbers x, x + 2, x + 6 where all three numbers arealso ludic numbers. Show all triplets of ludic numbers < 250 (Stretch goal)EOS (for/list ((x (in-range 250)) #:when (and (lucid? x) (lucid? (+ x 2)) (lucid? (+ x 6))))   (list x (+ x 2) (+ x 6))))`
Output:
```Generate and show here the first 25 ludic numbers.
(1 2 3 5 7 11 13 17 23 25 29 37 41 43 47 53 61 67 71 77 83 89 91 97 107)
How many ludic numbers are there less than or equal to 1000?
142
Show the 2000..2005'th ludic numbers.
(21481 21487 21493 21503 21511 21523)
A triplet is any three numbers x, x + 2, x + 6 where all three numbers are
also ludic numbers. Show all triplets of ludic numbers < 250 (Stretch goal)
((1 3 7) (5 7 11) (11 13 17) (23 25 29) (41 43 47) (173 175 179) (221 223 227) (233 235 239))
cpu time: 18753 real time: 18766 gc time: 80```

## REXX

`/*REXX program to display (a range of) ludic numbers, or a count of same*/parse arg N count bot top triples .    /*obtain optional parameters/args*/if N==''        then N=25              /*Not specified? Use the default.*/if count==''    then count=1000        /* "      "       "   "     "    */if bot==''      then bot=2000          /* "      "       "   "     "    */if top==''      then top=2005          /* "      "       "   "     "    */if triples==''  then triples=250-1     /* "      "       "   "     "    */say 'The first '   N   " ludic numbers: "   ludic(n)saysay "There are " words(ludic(-count)) ' ludic numbers from 1───►'count " (inclusive)."saysay "The "  bot   ' to '   top    " ludic numbers are: "    ludic(bot,top)\$=ludic(-triples) 0 0;     #=0;   @=say     do j=1  for words(\$); _=word(\$,j) /*it is known that ludic _ exists*/     if wordpos(_+2,\$)==0 | wordpos(_+6,\$)==0  then iterate   /*¬triple.*/     #=#+1;    @=@ '◄'_  _+2  _+6"► "  /*bump triple counter,  and ···  */     end   /*j*/                       /* [↑]  append found triple ──► @*/ if @==''  then  say  'From 1──►'triples", no triples found."          else  say  'From 1──►'triples", "   #   ' triples found:'   @exit                                   /*stick a fork in it, we're done.*//*──────────────────────────────────LUDIC subroutine────────────────────*/ludic: procedure; parse arg m 1 mm,h;  am=abs(m);     if h\==''  then am=h\$=1 2;    @=                           /*\$=ludic #s superset, @=# series*/                                       /* [↓]  construct a ludic series.*/  do j=3  by 2  to am * max(1,15*((m>0)|h\==''));  @=@ j;  end;     @=@' '                                       /* [↑]  high limit: approx|exact */  do  while  words(@)\==0              /* [↓]  examine the first word.  */  f=word(@,1);       \$=\$ f             /*append this first word to list.*/       do d=1  by f  while d<=words(@) /*use 1st #, elide all occurances*/       @=changestr(' 'word(@,d)" ",@, ' . ')  /*delete the # in the seq#*/       end   /*d*/                     /* [↑]  done eliding "1st" number*/  @=translate(@,,.)                    /*translate periods to blanks.   */  end         /*forever*/              /* [↑]  done eliding ludic #s.   */@=space(@)                             /*remove extra blanks from list. */ if h==''  then return subword(\$,1,am)  /*return a range of ludic numbers*/               return subword(\$,m,h-m+1)  /*return a section of a range.*/`

Some older REXXes don't have a changestr bif, so one is included here ──► CHANGESTR.REX.

output   using the defaults for input:

```The first  25  ludic numbers:  1 2 3 5 7 11 13 17 23 25 29 37 41 43 47 53 61 67 71 77 83 89 91 97 107

There are  142  ludic numbers from 1───►1000  (inclusive).

The  2000  to  2005  ludic numbers are:  21475 21481 21487 21493 21503 21511

From 1──►249,  8  triples found:  ◄1 3 7►  ◄5 7 11►  ◄11 13 17►  ◄23 25 29►  ◄41 43 47►  ◄173 175 179►  ◄221 223 227►  ◄233 235 239►
```

## Ruby

`def ludic(nmax=100000)  Enumerator.new do |y|    y << 1    ary = *2..nmax    until ary.empty?      y << (n = ary.first)      (0...ary.size).step(n){|i| ary[i] = nil}      ary.compact!    end  endend puts "First 25 Ludic numbers:", ludic.first(25).to_s puts "Ludics below 1000:", ludic(1000).count puts "Ludic numbers 2000 to 2005:", ludic.first(2005).last(6).to_s ludics = ludic(250).to_aputs "Ludic triples below 250:",     ludics.select{|x| ludics.include?(x+2) and ludics.include?(x+6)}.map{|x| [x, x+2, x+6]}.to_s`
Output:
```First 25 Ludic numbers:
[1, 2, 3, 5, 7, 11, 13, 17, 23, 25, 29, 37, 41, 43, 47, 53, 61, 67, 71, 77, 83, 89, 91, 97, 107]
Ludics below 1000:
142
Ludic numbers 2000 to 2005:
[21475, 21481, 21487, 21493, 21503, 21511]
Ludic triples below 250:
[[1, 3, 7], [5, 7, 11], [11, 13, 17], [23, 25, 29], [41, 43, 47], [173, 175, 179], [221, 223, 227], [233, 235, 239]]
```

## Seed7

`\$ include "seed7_05.s7i"; const func set of integer: ludicNumbers (in integer: n) is func  result    var set of integer: ludicNumbers is {1};  local    var set of integer: sieve is EMPTY_SET;    var integer: ludicNumber is 0;    var integer: number is 0;    var integer: count is 0;  begin    sieve := {2 .. n};    while sieve <> EMPTY_SET do      ludicNumber := min(sieve);      incl(ludicNumbers, ludicNumber);      count := 0;      for number range sieve do        if count rem ludicNumber = 0 then          excl(sieve, number);        end if;        incr(count);      end for;    end while;  end func; const integer: limit is 22000;const set of integer: ludicNumbers is ludicNumbers(limit); const proc: main is func  local    var integer: number is 0;    var integer: count is 0;  begin    write("First 25:");    for number range ludicNumbers until count = 25 do      write(" " <& number);      incr(count);    end for;    writeln;    count := 0;    for number range ludicNumbers until number > 1000 do      incr(count);    end for;    writeln("Ludics below 1000: " <& count);    write("Ludic 2000 to 2005:");    count := 0;    for number range ludicNumbers until count >= 2005 do      incr(count);      if count >= 2000 then        write(" " <& number);      end if;    end for;    writeln;    write("Triples below 250:");    for number range ludicNumbers until number > 250 do      if number + 2 in ludicNumbers and number + 6 in ludicNumbers then        write(" (" <& number <& ", " <& number + 2 <& ", " <& number + 6 <& ")");      end if;    end for;    writeln;  end func;`
Output:
```First 25: 1 2 3 5 7 11 13 17 23 25 29 37 41 43 47 53 61 67 71 77 83 89 91 97 107
Ludics below 1000: 142
Ludic 2000 to 2005: 21475 21481 21487 21493 21503 21511
Triples below 250: (1, 3, 7) (5, 7, 11) (11, 13, 17) (23, 25, 29) (41, 43, 47) (173, 175, 179) (221, 223, 227) (233, 235, 239)
```

## Tcl

Works with: Tcl version 8.6

The limit on the number of values generated is the depth of stack; this can be set to arbitrarily deep to go as far as you want. Provided you are prepared to wait for the values to be generated.

`package require Tcl 8.6 proc ludic n {    global ludicList ludicGenerator    for {} {[llength \$ludicList] <= \$n} {lappend ludicList \$i} {	set i [\$ludicGenerator]	set ludicGenerator [coroutine L_\$i apply {{gen k} {	    yield [info coroutine]	    while true {		set val [\$gen]		if {[incr i] == \$k} {set i 0} else {yield \$val}	    }	}} \$ludicGenerator \$i]    }    return [lindex \$ludicList \$n]}# Bootstrap the generator sequenceset ludicList [list 1]set ludicGenerator [coroutine L_1 apply {{} {    set n 1    yield [info coroutine]    while true {yield [incr n]}}}] # Default of 1000 is not enoughinterp recursionlimit {} 5000 for {set i 0;set l {}} {\$i < 25} {incr i} {lappend l [ludic \$i]}puts "first25: [join \$l ,]" for {set i 0} {[ludic \$i] <= 1000} {incr i} {}puts "below=1000: \$i" for {set i 1999;set l {}} {\$i < 2005} {incr i} {lappend l [ludic \$i]}puts "2000-2005: [join \$l ,]" for {set i 0} {[ludic \$i] < 256} {incr i} {set isl([ludic \$i]) \$i}for {set i 1;set l {}} {\$i < 250} {incr i} {    if {[info exists isl(\$i)] && [info exists isl([expr {\$i+2}])] && [info exists isl([expr {\$i+6}])]} {	lappend l (\$i,[expr {\$i+2}],[expr {\$i+6}])    }}puts "triplets: [join \$l ,]"`
Output:
```first25: 1,2,3,5,7,11,13,17,23,25,29,37,41,43,47,53,61,67,71,77,83,89,91,97,107
below=1000: 142
2000-2005: 21475,21481,21487,21493,21503,21511
triplets: (1,3,7),(5,7,11),(11,13,17),(23,25,29),(41,43,47),(173,175,179),(221,223,227),(233,235,239)
```

## VBScript

` Set list = CreateObject("System.Collections.Arraylist")Set ludic = CreateObject("System.Collections.Arraylist") 'populate the listFor i = 1 To 25000	list.Add iNext 'set 1 as the first ludic numberludic.Add list(0)list.RemoveAt(0) 'variable to count ludic numbers <= 1000up_to_1k = 1 'determine the succeeding ludic numbersFor j = 2 To 2005	If list.Count > 0 Then		If list(0) <= 1000 Then			up_to_1k = up_to_1k + 1		End If		ludic.Add list(0)	Else		Exit For	End If	increment = list(0) - 1	n = 0	Do While n <= list.Count - 1		list.RemoveAt(n)		n = n + increment	LoopNext 'the first 25 ludicsWScript.StdOut.WriteLine "First 25 Ludic Numbers:"For k = 0 To 24	If k < 24 Then		WScript.StdOut.Write ludic(k) & ", "	Else		WScript.StdOut.Write ludic(k)	End IfNextWScript.StdOut.WriteBlankLines(2) 'the number of ludics up to 1000WScript.StdOut.WriteLine "Ludics up to 1000: "WScript.StdOut.WriteLine up_to_1kWScript.StdOut.WriteBlankLines(1) '2000th - 2005th ludicsWScript.StdOut.WriteLine "The 2000th - 2005th Ludic Numbers:"For k = 1999 To 2004	If k < 2004 Then		WScript.StdOut.Write ludic(k) & ", "	Else		WScript.StdOut.Write ludic(k)	End IfNextWScript.StdOut.WriteBlankLines(2) 'triplets up to 250: x, x+2, and x+6WScript.StdOut.WriteLine "Ludic Triplets up to 250: "triplets = ""k = 0Do While ludic(k) + 6 <= 250	x2 = ludic(k) + 2	x6 = ludic(k) + 6	If ludic.IndexOf(x2,1) > 0 And ludic.IndexOf(x6,1) > 0 Then		triplets = triplets & ludic(k) & ", " & x2 & ", " & x6 & vbCrLf	End If	k = k + 1LoopWScript.StdOut.WriteLine triplets `
Output:
```First 25 Ludic Numbers:
1, 2, 3, 5, 7, 11, 13, 17, 23, 25, 29, 37, 41, 43, 47, 53, 61, 67, 71, 77, 83, 89, 91, 97, 107

Ludics up to 1000:
142

The 2000th - 2005th Ludic Numbers:
21475, 21481, 21487, 21493, 21503, 21511

Ludic Triplets up to 250:
1, 3, 7
5, 7, 11
11, 13, 17
23, 25, 29
41, 43, 47
173, 175, 179
221, 223, 227
233, 235, 239
```

## zkl

This solution builds a stack of iterators, one for each Ludic number, each extending the previous iterator. A "master" iterator sits atop the stack and provides the interface to the stack. When the next Ludic number is requested, a "pulse train" ripples down and up and down ... the stack as numbers are crossed off the list(s) (figure of speech, no numbers are cached).

`fcn dropNth(n,seq){   if(n==2) return(seq.tweak(fcn(n){ if(n.isEven) Void.Skip else n })); // uggg, special case   seq.tweak(fcn(n,skipper,w){ if(0==(w.idx+1)%skipper) Void.Skip else n }.fp1(n),	Void,Void,True);} fcn ludic{  //-->Walker   Walker(fcn(rw){ w:=rw.value; n:=w.next(); rw.set(dropNth(n,w)); n }.fp(Ref([2..]))).push(1);}`
`ludic().walk(25).toString(*).println();ludic().reduce(fcn(sum,n){ if(n<1000) return(sum+1); return(Void.Stop,sum); },0).println();ludic().drop(1999).walk(6).println();  // Ludic's between 2000 & 2005 ls:=ludic().filter(fcn(n){ (n<250) and True or Void.Stop  });  // Ludic's < 250ls.filter('wrap(n){ ls.holds(n+2) and ls.holds(n+6) }).apply(fcn(n){ T(n,n+2,n+6) }).println();`
Output:
```L(1,2,3,5,7,11,13,17,23,25,29,37,41,43,47,53,61,67,71,77,83,89,91,97,107)
142
L(21475,21481,21487,21493,21503,21511)
L(L(1,3,7),L(5,7,11),L(11,13,17),L(23,25,29),L(41,43,47),L(173,175,179),L(221,223,227),L(233,235,239))
```