Rare numbers

Rare 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.

Definitions and restrictions

Rare   numbers are positive integers   n   where:

•   n   is expressed in base ten
•   r   is the reverse of   n     (decimal digits)
•   n   must be non-palindromic   ( n   ≠   r)
•   (n+r)   is the   sum
•   (n-r)   is the   difference   and must be positive
•   the   sum   and the   difference   must be perfect squares

Task

•   find and show the first   5   rare   numbers
•   find and show the first   8   rare   numbers       (optional)
•   find and show more   rare   numbers                (stretch goal)

Show all output here, on this page.

References

•   an   OEIS   entry:   A035519          rare numbers.
•   an   OEIS   entry:   A059755   odd rare numbers.
•   planetmath entry:   rare numbers.     (some hints)
•   author's  website:   rare numbers   by Shyam Sunder Gupta.     (lots of hints and some observations).

REXX

All of the hints (properties of rare numbers) within Shyam Sunder Gupta's webpage have been incorporated in this REXX program.

The   interesting observations   (from the above webpage)   are being considered to be added here. <lang rexx>/*REXX program to calculate and display an specified amount of rare numbers. */ numeric digits 20; w= digits() + digits() % 3 /*ensure enough decimal digs for calcs.*/ parse arg many start . /*obtain optional argument from the CL.*/ if many== | many=="," then many= 5 /*Not specified? Then use the default.*/

@dr.=0; @dr.2= 1; @dr.5=1 ; @dr.8= 1; @dr.9= 1 /*rare # must have these digital roots.*/ @ps.=0; @ps.2= 1; @ps.3= 1; @ps.7= 1; @ps.8= 1 /*perfect squares must end in these.*/ @end.=0; @end.1=1; @end.4=1; @end.6=1; @end.9=1 /*rare # must not end in these digits.*/ @dif.=0; @dif.2=1; @dif.3=1; @dif.7=1; @dif.8=1; @dif.9=1 /* A─Q mustn't be these digs.*/ @noq.=0; @noq.0=1; @noq.1=1; @noq.4=1; @noq.5=1; @noq.6=1; @noq.9=1 /*A=8, Q mustn't be*/ @149.=0; @149.1=1; @149.4=1; @149.9=1 /*values for Z that need a even Y. */

1. = 0 /*the number of rare numbers (so far)*/

@n05.=0; do i= 1 to 9; if i==0 | i==5 then iterate; @n05.i= 1; end /*¬1 ¬5*/ @eve.=0; do i=-8 by 2 to 8; @eve.i=1; end /*define even " some are negative.*/ @odd.=0; do i=-9 by 2 to 9; @odd.i=1; end /* " odd " " " " */

                                                /*N=10, 'cause 1 dig #s are palindromic*/
   do n=10;  parse var  n  a 2 b 3  -2 p +1 q /*get 1st\2nd\penultimate\last digits. */
   if @end.q  then iterate                      /*rare numbers can't end in: 1 4 6 or 9*/
   if q==3    then iterate

      select                                    /*weed some integers based on 1st digit*/
      when a==q  then do
                      if a==2|a==8 then nop     /*if A = Q,   then A must be  2  or 8. */
                                   else iterate /*A  not two or eight?       Then skip.*/
                      if b\==p  then iterate    /*B  not equal to  P?        Then skip.*/
                      end
      when a==2  then do; if q\==2 then iterate /*A = 2?     Then  Q  must also be  2. */
                          if b\==p then iterate /*" " "      Then  B  must equal    P. */
                 end
      when a==4  then do
                      if q\==0   then iterate   /*if Q not equal to zero, then skip it.*/
                      _= b - p                  /*calculate difference between B and P.*/
                      if @eve._  then iterate   /*Positive not even?      Then skip it.*/
                      end
      when a==6  then do
                      if @n05.q  then iterate   /*Q  not a zero or five?  Then skip it.*/
                      _= b - p                  /*calculate difference between B and P.*/
                      if @eve._  then iterate
                      end
      when a==8  then do
                      if @noq.q  then iterate   /*Q  isn't one of 2, 3, 7, 8?  Skip it.*/
                        select
                        when q==2  then            if b+p\==9                then iterate
                        when q==3  then do; if b>p         then if b-p\== 7  then iterate

else if b

1 then if b+p\==11 then iterate else if b==0 then if b+p\== 1 then iterate end when q==8 then if b\==p then iterate otherwise nop end /*select*/ end /* [↓] A is an odd digit. */ otherwise n= n + 10**(length(n) - 1) - 1 /*bump N so next N starts with even dig*/ iterate /*Now, go and use the next value of N.*/ end /*select*/ _= a - q; if @dif._ then iterate /*diff of A─Q must be: 0, 1, 4, 5, or 6*/ r= reverse(n) /*obtain the reverse of the number N. */ if r>n then iterate /*Difference will be negative? Skip it*/ if n==r then iterate /*Palindromic? Then it can't be rare.*/ d= n-r; parse var d -2 y +1 z /*obtain the last 2 digs of difference.*/ if @ps.z then iterate /*Not 0, 1, 4, 5, 6, 9? Not perfect sq.*/ select when z==0 then if y\==0 then iterate /*Does Z = 0? Then Y must be zero. */ when z==5 then if y\==2 then iterate /*Does Z = 5? Then Y must be two. */ when z==6 then if y//2==0 then iterate /*Does Z = 6? Then Y must be odd. */ otherwise if @149.z then if y//2 then iterate /*Z=1,4,9? Y must be even*/ end /*select*/ s= n+r; parse var s -2 y +1 z /*obtain the last two digits of the sum*/ if @ps.z then iterate /*Not 0, 2, 5, 8, or 9? Not perfect sq.*/ select when z==0 then if y\==0 then iterate /*Does Z = 0? Then Y must be zero. */ when z==5 then if y\==2 then iterate /*Does Z = 5? Then Y must be two. */ when z==6 then if y//2==0 then iterate /*Does Z = 6? Then Y must be odd. */ otherwise if @149.z then if y//2 then iterate /*Z=1,4,9? Y must be even*/ end /*select*/ $= a + b /*a head start on figuring digital root*/ do k=3 for length(n) - 2 /*now, process the rest of the digits. */ $= $ + substr(n, k, 1) /*add the remainder of the digits in N.*/ end /*k*/ /*This REXX pgm uses 20 decimal digits.*/ do while $>9 /* [◄] Algorithm is good for 111 digs.*/ if $>9 then $= left($,1) + substr($,2,1)+ substr($,3,1,0) /*>9? Reduce to a dig*/ end /*while*/ if \@dr.$ then iterate /*Doesn't have good digital root? Skip*/ if iSqrt(s)**2 \== s then iterate /*Not a perfect square? Then skip it. */ if iSqrt(d)**2 \== d then iterate /* " " " " " " " */ #= # + 1 /*bump the counter of rare numbers. */ say right( th(#), length(#) + 9) ' rare number is: ' right( commas(n), w) if #>=many then leave /* [↑] W: the width of # with commas.*/ end /*n*/ exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ commas: parse arg _; do jc=length(_)-3 to 1 by -3; _=insert(',', _, jc); end; return _ th: parse arg th;return th||word('th st nd rd',1+(th//10)*(th//100%10\==1)*(th//10<4)) /*──────────────────────────────────────────────────────────────────────────────────────*/ iSqrt: parse arg x; $= 0; q= 1; do while q<=x; q= q*4 end /*while q<=x*/ do while q>1; q= q % 4; _= x-$-q; $= $ % 2 if _>=0 then do; x= _; $= $ + q end end /*while q>1*/; return $</lang>

output   when using the input of:     8

       1st  rare number is:                           65
       2nd  rare number is:                      621,770
       3rd  rare number is:                  281,089,082
       4th  rare number is:                2,022,652,202
       5th  rare number is:                2,042,832,002
       6th  rare number is:              868,591,084,757
       7th  rare number is:              872,546,974,178
       8th  rare number is:              872,568,754,178