3^1  = 2^1 + 1^1

5^2  = 4^2 + 3^2

6^3  = 5^3 + 4^3 + 3^3

15^4 = 14^4 + 9^4 + 8^4 + 6^4 + 4^4

12^5 = 4^5 + 5^5 + 6^5 + 7^5 + 9^5 + 11^5

25^6 = 1^6 + 2^6 + 3^6 + 5^6 + 6^6 + 7^6 + 8^6 + 9^6 + 10^6 + 12^6 + 13^6 + 15^6
     + 16^6 + 17^6 + 18^6 + 23^6
     
40^7 = 1^7 + 3^7 + 5^7 + 9^7 + 12^7 + 14^7 + 16^7 + 17^7 + 18^7 + 20^7 + 21^7
     + 22^7 + 25^7 + 28^7 + 39^7
     
84^8 = 1^8 + 2^8 + 3^8 + 5^8 + 7^8 + 9^8 + 10^8 + 11^8 + 12^8 + 13^8 + 14^8
     + 15^8 + 16^8 + 17^8 + 18^8 + 19^8 + 21^8 + 23^8 + 24^8 + 25^8 + 26^8 + 27^8
     + 29^8 + 32^8 + 33^8 + 35^8 + 37^8 + 38^8 + 39^8 + 41^8 + 42^8 + 43^8 + 45^8
     + 46^8 + 47^8 + 48^8 + 49^8 + 51^8 + 52^8 + 53^8 + 57^8 + 58^8 + 59^8 + 61^8
     + 63^8 + 69^8 + 73^8
       
47^9 = 1^9 + 2^9 + 4^9 + 7^9 + 11^9 + 14^9 + 15^9 + 18^9 + 26^9 + 27^9 + 30^9
     + 31^9 + 32^9 + 33^9 + 36^9 + 38^9 + 39^9 + 43^9
     
63^10 = 1^10 + 2^10 + 4^10 + 5^10 + 6^10 + 8^10 + 12^10 + 15^10 + 16^10 + 17^10
      + 20^10 + 21^10 + 25^10 + 26^10 + 27^10 + 28^10 + 30^10 + 36^10 + 37^10
      + 38^10 + 40^10 + 51^10 + 62^10
       
Comment from Jon Schoenfield, Nov 29 2008:
68^11 = 4^11 + 9^11 + 14^11 + 15^11 + 17^11 + 18^11 + 19^11 + 20^11 + 21^11
      + 22^11 + 24^11 + 27^11 + 28^11 + 30^11 + 31^11 + 32^11 + 33^11 + 36^11
      + 38^11 + 39^11 + 45^11 + 46^11 + 47^11 + 49^11 + 50^11 + 51^11 + 53^11
      + 54^11 + 55^11 + 65^11
        
81^12 = 1^12 + 5^12 + 7^12 + 8^12 + 9^12 + 10^12 + 20^12 + 21^12 + 25^12 + 28^12
      + 29^12 + 33^12 + 34^12 + 36^12 + 39^12 + 41^12 + 44^12 + 45^12 + 49^12
      + 51^12 + 54^12 + 55^12 + 57^12 + 61^12 + 66^12 + 68^12 + 70^12 + 77^12

Mathematica
102^13 = Total[{1, 3, 7, 9, 11, 13, 14, 15, 16, 18, 21, 23, 24, 26, 27, 30, 31,
       32, 33, 34, 36, 41, 42, 43, 44, 45, 46, 47, 48, 51, 52, 53, 55, 56, 57, 
       59, 60, 64, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 
       83, 86, 88, 89, 90}^13] (* Manol Iliev, Jan 04 2010 *)
       
95^14  = Total[{1, 2, 3, 5, 10, 12, 13, 14, 18, 19, 22, 24, 27, 28, 31, 34, 35,
       36, 37, 38, 39, 40, 41, 42, 43, 44, 48, 49, 51, 52, 53, 55, 57, 58, 59, 62, 
       63, 64, 65, 66, 70, 72, 73, 75, 77, 81, 84, 86, 87}^14] (* Manol Iliev, Apr 28 2011 *)
       
104^15 = Total[{2, 4, 6, 8, 10, 11, 15, 16, 20, 22, 23, 24, 26, 29, 32, 33, 35,
       39, 44, 47, 48, 49, 50, 54, 57, 62, 63, 65, 68, 70, 72, 75, 76, 77, 78,
       79, 80, 81, 83, 84, 85, 86, 87, 88, 95, 96}^15] (* Manol Iliev, Apr 28 2011 *)