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).


[ dup 1
    [ 2dup > while
      + 1 >>
      2dup / again ]
    drop nip ]                  is sqrt        (   n --> n )
 
[ dup sqrt 2 ** = not ]         is !square     (   n --> b )
 
[ number$ reverse 
  $->n drop ]                    is revnumber   (   n --> n )
 
[ 0 swap 
  [ base share /mod 
    rot + swap
    dup 0 = until ]
  drop ]                        is digitalroot (   n --> n )
 
 [ true swap
   dup revnumber
   2dup > not iff
     [ 2drop not ] done
   2dup + !square iff
     [ 2drop not ] done
   2dup - !square iff
     [ 2drop not ] done
   2drop ]                     is rare         (   n --> b )
 
  [ 0 
    [ 1+ dup rare if
        [ dup echo cr
          dip [ 1 - ] ]
      over 0 = until ]
    2drop ]                    is echorarenums (   n --> b )
 
5 echorarenums