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.

2, 4, 7, 12, 16, 23, 31

Offset: 0

Author

Pontus von Brömssen, Apr 10 2025

Keywords

• nonn
• hard
• more

Comments

Two such sets must have the same size, since the exponent m = 0 is allowed (with the usual convention that 0^0 = 1).

a(n) is the smallest k such that A382833(k,n) < 2^k.

Examples

			  n | a(n) | subsets with the same sums of powers
  --+------+-------------------------------------
  0 |   2  | {0}, {1}
  1 |   4  | {0,3}, {1,2}
  2 |   7  | {0,4,5}, {1,2,6}
  3 |  12  | {0,4,7,11}, {1,2,9,10}
  4 |  16  | {0,5,6,7,13,14}, {1,2,8,9,10,15}
  5 |  23  | {0,5,6,16,17,22}, {1,2,10,12,20,21}
  6 |  31  | {0,5,6,9,16,17,18,22,28,29}, {1,2,8,12,13,14,21,24,25,30}
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.
		

Crossrefs

Cf. A382382, A382833.