Gapful numbers

Gapful numbers is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

Numbers   (positive integers expressed in base ten)   that are (evenly) divisible by the number formed by the first and last digit are known as   gapful numbers.

All   one─   and two─digit   numbers have this property and are trivially excluded.   Only numbers   ≥ 100   will be considered for this Rosetta Code task.

Example

187   is a   gapful   number because it is evenly divisible by the number   17   which is formed by the first and last decimal digits of   187.

Task

•   Generate and show all sets of numbers (below) on one line (horizontally) with a title,   here on this page
•   Show the first   30   gapful numbers
•   Show the first   15   gapful numbers   ≥         1,000,000
•   Show the first   10   gapful numbers   ≥   1,000,000,000

Related task:

•   palindromic gapful numbers.

Also see

•   The OEIS entry:   A108343 gapful numbers.
•   numbersaplenty gapful numbers

Factor

<lang factor>USING: formatting kernel lists lists.lazy math math.functions math.text.utils sequences ;

gapful? ( n -- ? )

   dup 1 digit-groups [ first ] [ last 10 * + ] bi divisor? ;

30 100 15 1,000,000 10 1,000,000,000 [

   2dup lfrom [ gapful? ] lfilter ltake list>array
   "%d gapful numbers starting at %d:\n%[%d, %]\n\n" printf

] 2tri@</lang>

Output:

30 gapful numbers starting at 100:
{ 100, 105, 108, 110, 120, 121, 130, 132, 135, 140, 143, 150, 154, 160, 165, 170, 176, 180, 187, 190, 192, 195, 198, 200, 220, 225, 231, 240, 242, 253 }

15 gapful numbers starting at 1000000:
{ 1000000, 1000005, 1000008, 1000010, 1000016, 1000020, 1000021, 1000030, 1000032, 1000034, 1000035, 1000040, 1000050, 1000060, 1000065 }

10 gapful numbers starting at 1000000000:
{ 1000000000, 1000000001, 1000000005, 1000000008, 1000000010, 1000000016, 1000000020, 1000000027, 1000000030, 1000000032 }

Fōrmulæ

In this page you can see the solution of this task.

Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text (more info). Moreover, there can be multiple visual representations of the same program. Even though it is possible to have textual representation —i.e. XML, JSON— they are intended for transportation effects more than visualization and edition.

The option to show Fōrmulæ programs and their results is showing images. Unfortunately images cannot be uploaded in Rosetta Code.

Go

<lang go>package main

import "fmt"

func commatize(n uint64) string {

   s := fmt.Sprintf("%d", n)
   le := len(s)
   for i := le - 3; i >= 1; i -= 3 {
       s = s[0:i] + "," + s[i:]
   }
   return s

}

func main() {

   starts := []uint64{1e2, 1e6, 1e7, 1e9, 7123}
   counts := []int{30, 15, 15, 10, 25}
   for i := 0; i < len(starts); i++ {
       count := 0
       j := starts[i]
       pow := uint64(100)
       for {
           if j < pow*10 {
               break
           }
           pow *= 10
       }
       fmt.Printf("First %d gapful numbers starting at %s:\n", counts[i], commatize(starts[i]))
       for count < counts[i] {
           fl := (j/pow)*10 + (j % 10)
           if j%fl == 0 {
               fmt.Printf("%d ", j)
               count++
           }
           j++
           if j >= 10*pow {
               pow *= 10
           }
       }
       fmt.Println("\n")
   }

}</lang>

Output:

First 30 gapful numbers starting at 100:
100 105 108 110 120 121 130 132 135 140 143 150 154 160 165 170 176 180 187 190 192 195 198 200 220 225 231 240 242 253 

First 15 gapful numbers starting at 1,000,000:
1000000 1000005 1000008 1000010 1000016 1000020 1000021 1000030 1000032 1000034 1000035 1000040 1000050 1000060 1000065 

First 15 gapful numbers starting at 10,000,000:
10000000 10000001 10000003 10000004 10000005 10000008 10000010 10000016 10000020 10000030 10000032 10000035 10000040 10000050 10000060 

First 10 gapful numbers starting at 1,000,000,000:
1000000000 1000000001 1000000005 1000000008 1000000010 1000000016 1000000020 1000000027 1000000030 1000000032 

First 25 gapful numbers starting at 7,123:
7125 7140 7171 7189 7210 7272 7275 7280 7296 7350 7373 7420 7425 7474 7488 7490 7560 7575 7630 7632 7676 7700 7725 7770 7777 

Julia

<lang julia>using Lazy, Formatting

firstlast(a) = 10 * a[end] + a[1] isgapful(n) = (d = digits(n); length(d) < 3 || (m = firstlast(d)) != 0 && mod(n, m) == 0) gapfuls(start) = filter(isgapful, Lazy.range(start))

for (x, n) in [(100, 30), (1_000_000, 15), (1_000_000_000, 10)]

   println("First $n gapful numbers starting at ", format(x, commas=true), ":\n",
       take(n, gapfuls(x)))

end

</lang>

Output:

First 30 gapful numbers starting at 100:
(100 105 108 110 120 121 130 132 135 140 143 150 154 160 165 170 176 180 187 190 192 195 198 200 220 225 231 240 242 253)
First 15 gapful numbers starting at 1,000,000:
(1000000 1000005 1000008 1000010 1000016 1000020 1000021 1000030 1000032 1000034 1000035 1000040 1000050 1000060 1000065)
First 10 gapful numbers starting at 1,000,000,000:
(1000000000 1000000001 1000000005 1000000008 1000000010 1000000016 1000000020 1000000027 1000000030 1000000032)

Pascal

Translation of: Go

Works with: Free Pascal

Works with: Delphi

Now using using en passant updated MOD-values. Only recognizable for huge amounts of tests 100|74623687 ( up to 1 billion )-> takes 1.845s instead of 11.25s <lang pascal>program gapful; {$IFDEF FPC}

  {$MODE DELPHI}{$OPTIMIZATION ON,ALL}

{$ELSE}

 {$APPTYPE CONSOLE}

{$ENDIF}

uses

 sysutils,// IntToStr
 strUtils;// Numb2USA aka commatize

const

 cIdx = 5;
 starts : array [0..cIdx-1] of Uint64
        = (100,1000*1000, 10*1000*1000,1000*1000*1000, 7123);
 counts : array [0..cIdx-1] of Uint64 = (30, 15,15, 10, 25);
 //100|  74623687  =>    1000*1000*1000
 //100| 746236131  => 10*1000*1000*1000
 //100|7462360431  =>100*1000*1000*1000
 Base  = 10;

var

 ModsHL : array[0..99] of NativeUint;
 Pow10    : Uint64;    //global, seldom used
 countLmt : NativeUint;//Uint64; only for extreme counting

procedure OutHeader(i: NativeInt); Begin

 writeln('First ',counts[i],', gapful numbers starting at ',
          Numb2USA(IntToStr(starts[i])));

end;

procedure OutNum(n:Uint64); Begin

 write(' ',n);

end;

procedure InitMods(n:Uint64;H_dgt:NativeUint); //calculate first mod of n, when it reaches n var

 i,j : NativeInt;

Begin

 j := H_dgt; //= H_dgt+i
 For i := 0 to Base-1 do
 Begin
   ModsHL[j] := n MOD j;
   inc(n);
   inc(j);
 end;

end;

procedure InitMods2(n:Uint64;H_dgt,L_Dgt:NativeUint); //calculate first mod of n, when it reaches n //beware, that the lower n are reached in the next base round var

 i,j : NativeInt;

Begin

 j := H_dgt;
 n := n-L_Dgt;
 For i := 0 to L_Dgt-1 do
 Begin
   ModsHL[j] := (n+base) MOD j;
   inc(n);
   inc(j);
 end;
 For i := L_Dgt to Base-1 do
 Begin
   ModsHL[j] := n MOD j;
   inc(n);
   inc(j);
 end;

end;

procedure Main(TestNum:Uint64;Cnt:NativeUint); var

 LmtNextNewHiDgt: Uint64;
 tmp,LowDgt,GapNum : NativeUint;

Begin

 countLmt := cnt;
 Pow10 := Base*Base;
 LmtNextNewHiDgt := Base*Pow10;
 while LmtNextNewHiDgt <= TestNum do
 Begin
   Pow10 := LmtNextNewHiDgt;
   LmtNextNewHiDgt *= Base;
 end;
 LowDgt := TestNum MOD Base;
 GapNum  := TestNum DIV Pow10;
 LmtNextNewHiDgt := (GapNum+1)*Pow10;
 GapNum := Base*GapNum;
 IF LowDgt <> 0 then
   InitMods2(TestNum,GapNum,LowDgt)
 else
   InitMODS(TestNum,GapNum);

 GapNum += LowDgt;
 repeat

// if TestNum MOD (GapNum) = 0 then

   if ModsHL[GapNum]=0 then
   Begin
     tmp := countLmt-1;
     IF tmp < 32 then
       OutNum(TestNum);
     countLmt := tmp;
     // Test and BREAK only if something has changed
     IF tmp = 0 then
       BREAK;
   end;
   tmp := Base + ModsHL[GapNum];
   //translate into "if-less" version 3.35s -> 1.85s
   //bad branch prediction :-(
   //if tmp >= GapNum then tmp -= GapNum;
   tmp -= (-ORD(tmp >=GapNum) AND GapNum);
   ModsHL[GapNum]:= tmp;

   TestNum += 1;
   tmp := LowDgt+1;
   GapNum+=1;
   IF tmp >= Base then
   Begin
     tmp := 0;
     GapNum -= Base;
   end;
   LowDgt := tmp;
   //next Hi Digit
   if TestNum >= LmtNextNewHiDgt then
   Begin
     LowDgt := 0;
     GapNum +=Base;
     LmtNextNewHiDgt += Pow10;
     //next power of 10
     if GapNum >= Base*Base then
     Begin
       Pow10 *= Base;
       LmtNextNewHiDgt := 2*Pow10;
       GapNum := Base;
     end;
     initMods(TestNum,GapNum);
   end;
 until false;

end;

var

 i : integer;

Begin

 for i := 0 to High(starts) do
 Begin
   OutHeader(i);
   Main(starts[i],counts[i]);
   writeln(#13#10);
 end;

end.</lang>

Output:

First 30, gapful numbers starting at 100
 100 105 108 110 120 121 130 132 135 140 143 150 154 160 165 170 176 180 187 190 192 195 198 200 220 225 231 240 242 253

First 15, gapful numbers starting at 1,000,000
 1000000 1000005 1000008 1000010 1000016 1000020 1000021 1000030 1000032 1000034 1000035 1000040 1000050 1000060 1000065

First 15, gapful numbers starting at 10,000,000
 10000000 10000001 10000003 10000004 10000005 10000008 10000010 10000016 10000020 10000030 10000032 10000035 10000040 10000050 10000060

First 10, gapful numbers starting at 1,000,000,000
 1000000000 1000000001 1000000005 1000000008 1000000010 1000000016 1000000020 1000000027 1000000030 1000000032

First 25, gapful numbers starting at 7,123
 7125 7140 7171 7189 7210 7272 7275 7280 7296 7350 7373 7420 7425 7474 7488 7490 7560 7575 7630 7632 7676 7700 7725 7770 7777
_____
First 74623687, gapful numbers starting at 100
 999998976 999999000 999999090 999999091 999999099 999999152 999999165 999999180 999999270 999999287 999999333 999999354 999999355 999999360 999999448 999999450 999999456 999999540 999999545 999999612 999999630 999999720 999999735 999999810 999999824 999999900 999999925 999999936 999999938 999999990 1000000000

real    0m1,845s
start  |    count
  //100|  74623687  =>    1000*1000*1000
  //100| 746236131  => 10*1000*1000*1000
  //100|7462360431  =>100*1000*1000*1000 

Perl

<lang perl>use strict; use warnings; use feature 'say';

sub comma { reverse ((reverse shift) =~ s/(.{3})/$1,/gr) =~ s/^,//r }

sub is_gapful { my $n = shift; 0 == $n % join(, (split //, $n)[0,-1]) }

use constant Inf => 1e10; for ([1e2, 30], [1e6, 15], [1e9, 10], [7123, 25]) {

   my($start, $count) = @$_;
   printf "\nFirst $count gapful numbers starting at %s:\n", comma $start;
   my $n = 0; my $g = ;
   $g .= do { $n < $count ? (is_gapful($_) and ++$n and "$_ ") : last } for $start .. Inf;
   say $g;

}</lang>

Output:

First 30 gapful numbers starting at 100:
100 105 108 110 120 121 130 132 135 140 143 150 154 160 165 170 176 180 187 190 192 195 198 200 220 225 231 240 242 253

First 15 gapful numbers starting at 1,000,000:
1000000 1000005 1000008 1000010 1000016 1000020 1000021 1000030 1000032 1000034 1000035 1000040 1000050 1000060 1000065

First 10 gapful numbers starting at 1,000,000,000:
1000000000 1000000001 1000000005 1000000008 1000000010 1000000016 1000000020 1000000027 1000000030 1000000032

First 25 gapful numbers starting at 7,123:
7125 7140 7171 7189 7210 7272 7275 7280 7296 7350 7373 7420 7425 7474 7488 7490 7560 7575 7630 7632 7676 7700 7725 7770 7777

Perl 6

Works with: Rakudo version 2019.07.1

Also test starting on a number that doesn't start with 1. Required to have titles, may as well make 'em noble. :-)

<lang perl6>use Lingua::EN::Numbers;

for (1e2, 30, 1e6, 15, 1e9, 10, 7123, 25)».Int -> $start, $count {

   put "\nFirst $count gapful numbers starting at {comma $start}:\n" ~
   <Sir Lord Duke King>.pick ~ ": ", ~
   ($start..*).grep( { $_ %% .comb[0, *-1].join } )[^$count];

}</lang>

Output:

First 30 gapful numbers starting at 100:
Sir: 100 105 108 110 120 121 130 132 135 140 143 150 154 160 165 170 176 180 187 190 192 195 198 200 220 225 231 240 242 253

First 15 gapful numbers starting at 1,000,000:
Duke: 1000000 1000005 1000008 1000010 1000016 1000020 1000021 1000030 1000032 1000034 1000035 1000040 1000050 1000060 1000065

First 10 gapful numbers starting at 1,000,000,000:
King: 1000000000 1000000001 1000000005 1000000008 1000000010 1000000016 1000000020 1000000027 1000000030 1000000032

First 25 gapful numbers starting at 7,123:
King: 7125 7140 7171 7189 7210 7272 7275 7280 7296 7350 7373 7420 7425 7474 7488 7490 7560 7575 7630 7632 7676 7700 7725 7770 7777

Python

<lang python>from itertools import islice, count for start, n in [(100, 30), (1_000_000, 15), (1_000_000_000, 10)]:

   print(f"\nFirst {n} gapful numbers from {start:_}")
   print(list(islice(( x for x in count(start) 
                       if (x % (int(str(x)[0]) * 10 + (x % 10)) == 0) )
                     , n)))</lang>

Output:

First 30 gapful numbers from 100
[100, 105, 108, 110, 120, 121, 130, 132, 135, 140, 143, 150, 154, 160, 165, 170, 176, 180, 187, 190, 192, 195, 198, 200, 220, 225, 231, 240, 242, 253]

First 15 gapful numbers from 1_000_000
[1000000, 1000005, 1000008, 1000010, 1000016, 1000020, 1000021, 1000030, 1000032, 1000034, 1000035, 1000040, 1000050, 1000060, 1000065]

First 10 gapful numbers from 1_000_000_000
[1000000000, 1000000001, 1000000005, 1000000008, 1000000010, 1000000016, 1000000020, 1000000027, 1000000030, 1000000032]

REXX

<lang rexx>/*REXX program computes and displays gapful numbers and also palindromic gapful numbers.*/ numeric digits 20 /*ensure enough decimal digits gapfuls.*/ parse arg gapfuls /*obtain optional arguments from the CL*/ if gapfuls= then gapfuls= 30 25@7123 15@1000000 10@1000000000 /*assume defaults. */

       do until gapfuls=;      parse var gapfuls stuff gapfuls;       call gapful stuff
       end   /*until*/

exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ gapful: procedure; parse arg n '@' sp; #= 0; if sp== then sp= 100

       say center(' 'n      " gapful numbers starting at: "     sp' ', 125, "═")
       $=                                                          /*initialize $ list.*/
            do j=sp  until #==n                                    /*SP:  start point. */
            parse var   j   a  2    -1  b                        /*get 1st & last dig*/
            if j // (a||b) \== 0  then iterate                     /*perform ÷ into  J.*/
            #= # + 1;             $= $ j                           /*bump #; append──►$*/
            end   /*j*/
       say strip($);     say;     return</lang>

output   when using the default inputs:

(Shown at   ⁵/₆   size.)

═══════════════════════════════════════════ 30  gapful numbers starting at:  100 ════════════════════════════════════════════
100 105 108 110 120 121 130 132 135 140 143 150 154 160 165 170 176 180 187 190 192 195 198 200 220 225 231 240 242 253

═══════════════════════════════════════════ 25  gapful numbers starting at:  7123 ═══════════════════════════════════════════
7125 7140 7171 7189 7210 7272 7275 7280 7296 7350 7373 7420 7425 7474 7488 7490 7560 7575 7630 7632 7676 7700 7725 7770 7777

═════════════════════════════════════════ 15  gapful numbers starting at:  1000000 ══════════════════════════════════════════
1000000 1000005 1000008 1000010 1000016 1000020 1000021 1000030 1000032 1000034 1000035 1000040 1000050 1000060 1000065

════════════════════════════════════════ 10  gapful numbers starting at:  1000000000 ════════════════════════════════════════
1000000000 1000000001 1000000005 1000000008 1000000010 1000000016 1000000020 1000000027 1000000030 1000000032

zkl

<lang zkl>fcn gapfulW(start){ //--> iterator

  [start..].tweak(
     fcn(n){ if(n % (10*n.toString()[0] + n%10)) Void.Skip else n })

}</lang> <lang zkl>foreach n,z in

      ( T( T(100, 30), T(1_000_000, 15), T(1_000_000_000, 10), T(7_123,25) )){
  println("First %d gapful numbers starting at %,d:".fmt(z,n));
  gapfulW(n).walk(z).concat(", ").println("\n");

}</lang>

Output:

First 30 gapful numbers starting at 100:
100, 105, 108, 110, 120, 121, 130, 132, 135, 140, 143, 150, 154, 160, 165, 170, 176, 180, 187, 190, 192, 195, 198, 200, 220, 225, 231, 240, 242, 253

First 15 gapful numbers starting at 1,000,000:
1000000, 1000005, 1000008, 1000010, 1000016, 1000020, 1000021, 1000030, 1000032, 1000034, 1000035, 1000040, 1000050, 1000060, 1000065

First 10 gapful numbers starting at 1,000,000,000:
1000000000, 1000000001, 1000000005, 1000000008, 1000000010, 1000000016, 1000000020, 1000000027, 1000000030, 1000000032

First 25 gapful numbers starting at 7,123:
7125, 7140, 7171, 7189, 7210, 7272, 7275, 7280, 7296, 7350, 7373, 7420, 7425, 7474, 7488, 7490, 7560, 7575, 7630, 7632, 7676, 7700, 7725, 7770, 7777