A146756 - OEIS

%I #29 Dec 22 2021 02:58:30

%S 1,65,5104,300834,9006349824,82188309244

%N a(n) is the smallest number expressible as the sum of n distinct positive n-th powers in exactly n ways.

%C It is not known if any terms exist in this sequence beyond n = 6.

%C Per email communication from Ian Rayson, the n-th powers for each sum must be distinct. If duplicate n-th powers were allowed, a(4) would be 236674 while the other terms would remain unchanged. - _Donovan Johnson_, Nov 08 2008

%C If duplicate n-th powers were allowed, a(2) would be 50. - _Jonathan Sondow_, Oct 23 2013

%e a(2) = 65 = 1^2 + 8^2 = 4^2 + 7^2.

%e a(5) = 9006349824 = 8^5 + 34^5 + 62^5 + 68^5 + 92^5 = 8^5 + 41^5 + 47^5 + 79^5 + 89^5 = 12^5 + 18^5 + 72^5 + 78^5 + 84^5 = 21^5 + 34^5 + 43^5 + 74^5 + 92^5 = 24^5 + 42^5 + 48^5 + 54^5 + 96^5. - _Donovan Johnson_, Nov 08 2008

%e From _Michael S. Branicky_, Dec 21 2021: (Start)

%e a(6) = 82188309244 =  1^6 +  9^6 + 29^6 + 44^6 + 55^6 + 60^6,

%e                    =  2^6 + 12^6 + 25^6 + 51^6 + 53^6 + 59^6,

%e                    =  5^6 + 23^6 + 27^6 + 44^6 + 51^6 + 62^6,

%e                    = 10^6 + 16^6 + 41^6 + 45^6 + 51^6 + 61^6,

%e                    = 12^6 + 23^6 + 33^6 + 34^6 + 55^6 + 61^6,

%e                    = 15^6 + 23^6 + 31^6 + 36^6 + 53^6 + 62^6. (End)

%Y Cf. A230477 (same except that the n-th powers need not be distinct and the number of ways is at least n, not necessarily exactly n). - _Jonathan Sondow_, Oct 23 2013

%K nonn,hard,more

%O 1,2

%A Ian Rayson (Ian.Rayson(AT)studentmail.newcastle.edu.au), Nov 02 2008

%E a(5) from _Donovan Johnson_, Nov 08 2008

%E Definition clarified by _Jonathan Sondow_, Oct 23 2013

%E a(6) from _Michael S. Branicky_, May 09 2021