A230718 Smallest n-th power equal to a sum of some consecutive, immediately preceding, positive n-th powers, or 0 if none.

1, 3, 25, 216, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Author

Jonathan Sondow, Oct 28 2013

Keywords

• nonn

Comments

a(n) is the smallest solution to k^n + (k+1)^n + ... + (k+m)^n = (k+m+1)^n with k > 0 and m > 0, or 0 if none.

Dickson says Escott proved that for 2 <= n <= 5, the only solutions are 3^2 + 4^2 = 5^2 and 3^3 + 4^3 + 5^3 = 6^3. Thus a(4) = a(5) = 0.

Is a(n) != 0 for any n > 3?

The Erdos-Moser equation is the case k = 1. They conjecture that the only solution is m = n = 1. Any counterexample would be a case of a(n) > 0 with n > 3. And such a case with k = 1 would be a counterexample to the Erdos-Moser conjecture.

Examples

			1^0 = 2^0 = 1.
1^1 + 2^1 = 3^1 = 3.
3^2 + 4^2 = 5^2 = 25.
3^3 + 4^3 + 5^3 = 6^3 = 216.
		

References

• Ian Stewart, "Game, Set and Math", Dover, 2007, Chapter 8 'Close Encounters of the Fermat Kind', pp. 107-124.

Links

• L. E. Dickson, History of the Theory of Numbers, vol II, p. 585.

Extensions

More terms from Jinyuan Wang, Dec 31 2021