A230477 Smallest number that is the sum of n positive n-th powers in >= n ways.

1, 50, 5104, 236674, 9006349824, 82188309244

Offset

1,2

Comments

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?

References

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

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

Links

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

Open Directory Project, Equal Sums of Like Powers

Formula

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

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

Example

1 = 1^1.
50 = 1^2 + 7^2 = 5^2 + 5^2.
5104 = 1^3 + 12^3 + 15^3 = 2^3 + 10^3 + 16^3 = 9^3 + 10^3 + 15^3.
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.
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.
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.

Crossrefs

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

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).

Sequence in context: A248205 A174756 A231937 * A320099 A115436 A184555

Adjacent sequences: A230474 A230475 A230476 * A230478 A230479 A230480

Keyword

nonn,more,hard

Author

Jonathan Sondow, Oct 22 2013

Extensions

a(5) from Donovan Johnson, Oct 23 2013

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

Status

approved