A146756 a(n) is the smallest number expressible as the sum of n distinct positive n-th powers in exactly n ways.

1, 65, 5104, 300834, 9006349824, 82188309244

Offset: 1

Author

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

Keywords

• nonn
• hard
• more

Comments

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

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

If duplicate n-th powers were allowed, a(2) would be 50. - Jonathan Sondow, Oct 23 2013

Examples

			a(2) = 65 = 1^2 + 8^2 = 4^2 + 7^2.
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
From _Michael S. Branicky_, Dec 21 2021: (Start)
a(6) = 82188309244 =  1^6 +  9^6 + 29^6 + 44^6 + 55^6 + 60^6,
                   =  2^6 + 12^6 + 25^6 + 51^6 + 53^6 + 59^6,
                   =  5^6 + 23^6 + 27^6 + 44^6 + 51^6 + 62^6,
                   = 10^6 + 16^6 + 41^6 + 45^6 + 51^6 + 61^6,
                   = 12^6 + 23^6 + 33^6 + 34^6 + 55^6 + 61^6,
                   = 15^6 + 23^6 + 31^6 + 36^6 + 53^6 + 62^6. (End)
		

Crossrefs

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

Extensions

a(5) from Donovan Johnson, Nov 08 2008

Definition clarified by Jonathan Sondow, Oct 23 2013

a(6) from Michael S. Branicky, May 09 2021