Palindromic gapful numbers
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
1037 is a gapful number because it is evenly divisible by the number 17 which is formed by the first and last decimal digits of 1037.
A palindromic number is (for this task, a positive integer expressed in base ten), when the number is
reversed, is the same as the original number.
- Task
-
- Show (nine sets) the first 20 palindromic gapful numbers that end with:
- the digit 1
- the digit 2
- the digit 3
- ··· ···
- the digit 9
- Show (nine sets, like above) of palindromic gapful numbers:
- the last 15 palindromic gapful numbers (out of 100)
- the last 10 palindromic gapful numbers (out of 1,000) {optional}
- Related tasks
- Also see
-
- The OEIS entry: A108343 gapful numbers.
REXX
<lang rexx>/*REXX program computes and displays palindromic gapful numbers, it also can show those */ /*─────────────────────── palindromic gapful numbers listed by their last decimal digit.*/ numeric digits 20 /*ensure enough decimal digits gapfuls.*/ parse arg pangaps if pangaps= then pangaps= 20 100@@15 1000@@10 /*assume some ranges*/
do until pangaps=; parse var pangaps stuff pangaps; call pangap stuff
end /*until*/
exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ pangap: procedure; parse arg n '@' sp '@' z; #= 0; if sp== then sp= 100
if z== then z= n
@which= ' last '; if z==n then @which= " first "
@pangap#Start= ' palindromic gapful numbers starting at: '
say center(@which z " of " n @pangap#Start sp" ", 140, "~")
#.= 0 /*array of result counts for each digit*/
tot= n * 9 /*total # of results that are wanted. */
$.=; sum= 0 /*blank lists; digit results (so far).*/
do j=sp until sum==tot /*loop 'til all digit counters filled. */
parse var j a 2 -1 b /*obtain the first and last dec. digit.*/
if a \==b then iterate /*Not a palindrome? Then skip it.*/
if #.b ==n then iterate /*Digit quota filled? " " " */
if reverse(j) \==j then iterate /*Not a palindrome? " " " */
if j // (a||b) \==0 then iterate /*Not divisible by A||B? " " " */
sum= sum + 1; #.b= #.b + 1; /*bump the sum counter & digit counter.*/
$.b= $.b j /*append J to the correct list. */
end /*j*/
/* [↓] just show the last Z numbers.*/
do k=1 for 9; say k':' strip( subword($.k, 1 + n - z) )
end /*k*/; say; return</lang>
- output when using the internal default inputs:
(Shown at 5/6 size.)
═════════════════════════════════════ first 20 of 20 palindromic gapful numbers starting at: 100 ══════════════════════════════════════ 1: 121 1001 1111 1221 1331 1441 1551 1661 1771 1881 1991 10901 11011 12221 13431 14641 15851 17171 18381 19591 2: 242 2002 2112 2222 2332 2442 2552 2662 2772 2882 2992 20702 21912 22022 23232 24442 25652 26862 28182 29392 3: 363 3003 3333 3663 3993 31713 33033 36663 300003 303303 306603 309903 312213 315513 318813 321123 324423 327723 330033 333333 4: 484 4004 4224 4444 4664 4884 40304 42724 44044 46464 48884 400004 401104 402204 403304 404404 405504 406604 407704 408804 5: 5005 5115 5225 5335 5445 5555 5665 5775 5885 5995 50105 51315 52525 53735 54945 55055 56265 57475 58685 59895 6: 6006 6336 6666 6996 61116 64746 66066 69696 600006 603306 606606 609906 612216 615516 618816 621126 624426 627726 630036 633336 7: 7007 7777 77077 700007 707707 710017 717717 720027 727727 730037 737737 740047 747747 750057 757757 760067 767767 770077 777777 780087 8: 8008 8448 8888 80608 86768 88088 800008 802208 804408 806608 808808 821128 823328 825528 827728 829928 840048 842248 844448 846648 9: 9009 9999 94149 99099 900009 909909 918819 927729 936639 945549 954459 963369 972279 981189 990099 999999 9459549 9508059 9557559 9606069 ═════════════════════════════════════ last 15 of 100 palindromic gapful numbers starting at: 100 ══════════════════════════════════════ 1: 165561 166661 167761 168861 169961 170071 171171 172271 173371 174471 175571 176671 177771 178871 179971 2: 265562 266662 267762 268862 269962 270072 271172 272272 273372 274472 275572 276672 277772 278872 279972 3: 30366303 30399303 30422403 30455403 30488403 30511503 30544503 30577503 30600603 30633603 30666603 30699603 30722703 30755703 30788703 4: 4473744 4485844 4497944 4607064 4619164 4620264 4632364 4644464 4656564 4668664 4681864 4693964 4803084 4815184 4827284 5: 565565 566665 567765 568865 569965 570075 571175 572275 573375 574475 575575 576675 577775 578875 579975 6: 60399306 60422406 60455406 60488406 60511506 60544506 60577506 60600606 60633606 60666606 60699606 60722706 60755706 60788706 60811806 7: 72299227 72322327 72399327 72422427 72499427 72522527 72599527 72622627 72699627 72722727 72799727 72822827 72899827 72922927 72999927 8: 80611608 80622608 80633608 80644608 80655608 80666608 80677608 80688608 80699608 80800808 80811808 80822808 80833808 80844808 80855808 9: 95311359 95400459 95499459 95588559 95677659 95766759 95855859 95944959 96033069 96122169 96211269 96300369 96399369 96488469 96577569 ═════════════════════════════════════ last 10 of 1000 palindromic gapful numbers starting at: 100 ═════════════════════════════════════ 1: 17799771 17800871 17811871 17822871 17833871 17844871 17855871 17866871 17877871 17888871 2: 27799772 27800872 27811872 27822872 27833872 27844872 27855872 27866872 27877872 27888872 3: 3084004803 3084334803 3084664803 3084994803 3085225803 3085555803 3085885803 3086116803 3086446803 3086776803 4: 482282284 482414284 482535284 482656284 482777284 482898284 482909284 483020384 483141384 483262384 5: 57800875 57811875 57822875 57833875 57844875 57855875 57866875 57877875 57888875 57899875 6: 6084004806 6084334806 6084664806 6084994806 6085225806 6085555806 6085885806 6086116806 6086446806 6086776806 7: 7452992547 7453223547 7453993547 7454224547 7454994547 7455225547 7455995547 7456226547 7456996547 7457227547 8: 8085995808 8086006808 8086116808 8086226808 8086336808 8086446808 8086556808 8086666808 8086776808 8086886808 9: 9675005769 9675995769 9676886769 9677777769 9678668769 9679559769 9680440869 9681331869 9682222869 9683113869