Equality in the sum of digits
Jelly, 13 bytes

,²DS€=/
1Ç³#Ṫ


Input is 1-indexed. Try it online!

How it works

1Ç³#Ṫ    Main link. Argument: n (index)

1        Set the return value to 1.
   #     Execute ... until ... matches have been found.
 Ç         the helper link
  ³        n
    Ṫ    Extract the last match.


,²DS€=/  Helper link. Argument: k (integer)

,²       Pair k with k².
  D      Convert each to decimal.
   S€    Compute the sum of each list of base 10 digits.
     =/  Reduce by equality.