Unprimeable numbers
As used here, all unprimeable numbers (positive integers) are always expressed in base ten.
- Definitions
───── Definition from OEIS ─────:
Unprimeable numbers are composite numbers that always remain composite when a single decimal digit of the number is changed.
───── Definition from Wiktionary (referenced from Adam Spencer's book) ─────:
(arithmetic) that cannot be turned into a prime number by changing just one of its digits to any other
digit.. (sic)
Unprimeable numbers are also spelled: unprimable.
All one─ and two─digit numbers can be turned into primes by changing a single decimal digit.
- Examples
190 isn't unprimeable, because by changing the zero digit into a three yields 193, which is a prime.
The number 200 is unprimeable, since none of the numbers 201, 202, 203, ··· 209 are prime, and all the other numbers obtained by changing a single digit to produce 100, 300, 400, ··· 900, or 210, 220, 230, ··· 290 which are all even.
It is valid to change 189 into 089 by changing the 1 (one) into
a 0 (zero), which then the leading zero can be removed, and then treated as if
the "new" number is 89.
- Task
-
- show the first 25 unprimeable numbers (horizontally, on one line, preferably with a title)
- show the 600th unprimeable number
- (optional) show the lowest unprimeable number ending in a specific decimal digit (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)
- (optional) use commas in the numbers where appropriate
Show all output here, on this page.
- Also see
-
- the OEIS entry: A118118 (unprimeable)
- the Wiktionary entry (reference from below): (arithmetic definition) unprimeable
- from the Adam Spencer book (page 200): Adam Spencer's World of Numbers (Xoum Publishing)
REXX
<lang rexx>/*REXX program finds and displays unprimeable numbers (non-negative integers). */ parse arg n x hp . /*obtain optional arguments from the CL*/ if n== | n=="," then n= 35 /*Not specified? Then use the default.*/ if x== | x=="," then x= 600 /* " " " " " " */ if hp== | hp=="," then hp= 10000000 /* " " " " " " */ eds=4; ed.1= 1; ed.2= 3; ed.3= 7; ed.4= 9 /*the "end" digits which are prime; #>9*/ call genP hp /*generate primes up to & including HP.*/
- = 0 /*number of unprimeable numbers so far.*/
$$=; $.=. /*a list " " " " " */
/*1─ and 2─digit #'s are all primeable.*/
do j=100; if !.j then iterate /*Prime? Unprimeable must be composite*/
L= length(j) /*obtain the length of the number J. */
meat= left(j, L-1) /*obtain the first L-1 digits of J. */
/* [↑] examine the "end" digit of J. */
do e_=1 for eds; new= meat || ed.e_ /*obtain a different number (than J).*/
if new==j then iterate /*Is it the original number? Then skip.*/
if !.new then iterate j /*This new number not prime? " " */
end /*e_*/
meat= right(j, L-1) /*obtain the last L-1 digits of J. */
/* [↑] examine a new 1st digit of J. */
do f_=0 for 10; new= (f_||meat) + 0 /*obtain a different number (than J).*/
if new==j then iterate /*Is it the original number? Then skip.*/
if !.new then iterate j /*This new number not prime? " " */
end /*f_*/ /* [↑] examine the front digit of J. */
do a_= 2 to L-1 /*traipse through the middle digits. */
meat= left(j, a_ - 1) /*use a number of left─most dec. digits*/
rest= substr(j, a_ + 1) /* " " " " right─most " " */
do n_=0 for 10 /*traipse through all 1─digit numbers. */
new= meat || n_ || rest /*construct new number, like a phoenix.*/
if new==j then iterate /*Is it the original number? Then skip.*/
if !.new then iterate j /*This new number not prime? " " */
end /*n_*/
end /*a_*/
#= # + 1 /*bump the count of unprimeable numbers*/
if #<n then $$= $$ j /*maybe add unprimeable # to $$ list.*/
if #==x then $.ox= j /*assign the Xth unprimeable number.*/
_= right(j, 1) /*obtain the right─most dec digit of J.*/
if $._==. then $._= j /*the 1st unprimeable # that ends in _.*/
if $.3==. then iterate; if $.7==. then iterate /*test if specific #'s found.*/
if $.1==. then iterate; if $.9==. then iterate /* " " " " " */
leave /*if here, then we're done. */
end /*j*/
if n>0 then do; say center(' first ' n "unprimeable numbers ", 135, '═')
say strip($$); say
end
if x>0 then say ' the ' th(x) " unprimeable number is: " commas($.ox) say
do o=0 for 10; if length($.o)==0 then iterate
say ' the first unprimeable number that ends in ' o " is:"right(commas($.o),11)
end /*o*/
exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ commas: parse arg ?; do c=length(?)-3 to 1 by -3; ?=insert(',', ?, c); end; return ? th:procedure;parse arg x;return x||word('th st nd rd',1+(x//10)*(x//100%10\==1)*(x//10<4)) /*──────────────────────────────────────────────────────────────────────────────────────*/ genP: @.1=2; @.2=3; @.3=5; @.4=7; @.5=11; @.6= 13; nP=6 /*assign low primes; # primes. */
!.=0; !.2=1; !.3=1; !.5=1; !.7=1; !.11=1 /*assign some low semaphore primes. */
do lim=100 until lim*lim>=hp; end /*only keep primes up to the sqrt(hp). */
do j=@.nP+4 by 2 to hp /*only find odd primes from here on. */
if j// 3==0 then iterate /*is J divisible by 3? Then not prime.*/
parse var j -1 _;if _==5 then iterate /*Is last digit a "5"? " " " */
if j// 7==0 then iterate /*is J divisible by 7? Then not prime.*/
if j//11==0 then iterate /*is J divisible by 11? Then not prime.*/
if j//13==0 then iterate /*is J divisible by 13? Then not prime.*/
do k=7 while k*k<=j /*divide by some known low odd primes. */
if j // @.k==0 then iterate j /*Is J divisible by P? Then not prime.*/
end /*k*/ /* [↓] a prime (J) has been found. */
nP= nP+1; if nP<=lim then @.nP=j; !.j=1 /*bump prime count; assign prime to @.*/
end /*j*/; return</lang>
- output when using the default inputs:
(Shown at 5/6 size.)
════════════════════════════════════════════════════ first 35 unprimeable numbers ════════════════════════════════════════════════════
200 204 206 208 320 322 324 325 326 328 510 512 514 515 516 518 530 532 534 535 536 538 620 622 624 625 626 628 840 842 844 845 846 848
the 600th unprimeable number is: 5,242
the first unprimeable number that ends in 0 is: 200
the first unprimeable number that ends in 1 is: 595,631
the first unprimeable number that ends in 2 is: 322
the first unprimeable number that ends in 3 is: 1,203,623
the first unprimeable number that ends in 4 is: 204
the first unprimeable number that ends in 5 is: 325
the first unprimeable number that ends in 6 is: 206
the first unprimeable number that ends in 7 is: 872,897
the first unprimeable number that ends in 8 is: 208
the first unprimeable number that ends in 9 is: 212,159