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A146756
a(n) is the smallest number expressible as the sum of n distinct positive n-th powers in exactly n ways.
13
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
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
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