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

1,2

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

Links

Table of n, a(n) for n=1..6.

Example

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

Sequence in context: A027535 A110900 A084272 * A202386 A294955 A115432

Adjacent sequences: A146753 A146754 A146755 * A146757 A146758 A146759

Keyword

nonn,hard,more

Author

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

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

Status

approved