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%I #14 Jan 12 2022 11:49:37
%S 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,
%T 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,
%U 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N Smallest n-th power equal to a sum of some consecutive, immediately preceding, positive n-th powers, or 0 if none.
%C 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.
%C 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.
%C Is a(n) != 0 for any n > 3?
%C 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.
%D Ian Stewart, "Game, Set and Math", Dover, 2007, Chapter 8 'Close Encounters of the Fermat Kind', pp. 107-124.
%H L. E. Dickson, <a href="https://openlibrary.org/books/OL6616242M/History_of_the_theory_of_numbers_...">History of the Theory of Numbers</a>, vol II, p. 585.
%e 1^0 = 2^0 = 1.
%e 1^1 + 2^1 = 3^1 = 3.
%e 3^2 + 4^2 = 5^2 = 25.
%e 3^3 + 4^3 + 5^3 = 6^3 = 216.
%K nonn
%O 0,2
%A _Jonathan Sondow_, Oct 28 2013
%E More terms from _Jinyuan Wang_, Dec 31 2021
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