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A382832 Least k such that there exist two distinct subsets of {0, ..., k-1} with the same sum of m-th powers for 0 <= m <= n.

%I A382832 #9 Apr 12 2025 09:42:45
%S A382832 2,4,7,12,16,23,31
%N A382832 Least k such that there exist two distinct subsets of {0, ..., k-1} with the same sum of m-th powers for 0 <= m <= n.
%C A382832 Two such sets must have the same size, since the exponent m = 0 is allowed (with the usual convention that 0^0 = 1).
%C A382832 a(n) is the smallest k such that A382833(k,n) < 2^k.
%e A382832   n | a(n) | subsets with the same sums of powers
%e A382832   --+------+-------------------------------------
%e A382832   0 |   2  | {0}, {1}
%e A382832   1 |   4  | {0,3}, {1,2}
%e A382832   2 |   7  | {0,4,5}, {1,2,6}
%e A382832   3 |  12  | {0,4,7,11}, {1,2,9,10}
%e A382832   4 |  16  | {0,5,6,7,13,14}, {1,2,8,9,10,15}
%e A382832   5 |  23  | {0,5,6,16,17,22}, {1,2,10,12,20,21}
%e A382832   6 |  31  | {0,5,6,9,16,17,18,22,28,29}, {1,2,8,12,13,14,21,24,25,30}
%e A382832 For n = 3, the two subsets {0,4,7,11} and {1,2,9,10} of {0, ..., 11} have the same sum of m-th powers for 0 <= m <= 3: 0^0+4^0+7^0+11^0 = 1^0+2^0+9^0+10^0 = 4, 0^1+4^1+7^1+11^1 = 1^1+2^1+9^1+10^1 = 22, 0^2+4^2+7^2+11^2 = 1^2+2^2+9^2+10^2 = 186, 0^3+4^3+7^3+11^3 = 1^3+2^3+9^3+10^3 = 1738. There are no such subsets of {0, ..., 10}, so a(3) = 12.
%Y A382832 Cf. A382382, A382833.
%K A382832 nonn,hard,more
%O A382832 0,1
%A A382832 _Pontus von Brömssen_, Apr 10 2025