A230477 - OEIS

%I #29 May 09 2021 21:48:07

%S 1,50,5104,236674,9006349824,82188309244

%N Smallest number that is the sum of n positive n-th powers in >= n ways.

%C Does a(6) exist? For which values of n does a(n) exist? Is there a proof that a(n) < a(n+1) when both exist?

%D A. H. Beiler, Recreations in the Theory of Numbers: The Queen of Mathematics Entertains, Dover, NY, 1966, pp. 162-165, 290-291.

%D R. K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer, 2004, D1.

%H Open Directory Project, <a href="http://www.dmoz.org/Science/Math/Number_Theory/Diophantine_Equations/Equal_Sums_of_Like_Powers/">Equal Sums of Like Powers</a>

%F a(n) <= A146756(n), with equality at least for n = 1, 3, 5 and inequality at least for n = 2, 4.

%F a(n) >= A091414(n) for n > 1, with equality at least for n = 2 and inequality at least for n = 3, 4, 5.

%e 1 = 1^1.

%e 50 = 1^2 + 7^2 = 5^2 + 5^2.

%e 5104 = 1^3 + 12^3 + 15^3 = 2^3 + 10^3 + 16^3 = 9^3 + 10^3 + 15^3.

%e 236674 = 1^4 + 2^4 + 7^4 + 22^4 = 3^4 + 6^4 + 18^4 + 19^4 = 7^4 + 14^4 + 16^4 + 19^4 = 8^4 + 16^4 + 17^4 + 17^4.

%e 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.

%e 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.

%Y a(2) = A048610(2), a(3) = A025398(1), a(4) = A219921(1).

%Y Cf. A146756 (smallest number that is the sum of n distinct positive n-th powers in exactly n ways), A230561 (smallest number that is the sum of two positive n-th powers in >= n ways), A091414 (smallest number that is the sum of n positive n-th powers in >= 2 ways).

%K nonn,more,hard

%O 1,2

%A _Jonathan Sondow_, Oct 22 2013

%E a(5) from _Donovan Johnson_, Oct 23 2013

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