A378623 Number of pairs (u,v) for which there is a set of n positive integers X = {x_1, ..., x_n} with u = |X+X| = #{x_i + x_j : 1 <= i,j <=n} and v = |X*X| = #{x_i * x_j : 1 <= i,j <=n}.

1, 1, 3, 10, 27, 69

Offset: 1

Author

Kevin O'Bryant, Dec 02 2024

Keywords

• nonn
• more
• hard

Comments

As 2n-1 <= |X+X|,|X*X| <= n(n+1)/2 by combinatorics, we know that a(n) <= 1/4 (n^2-3n+4)^2 <= 0.25 n^4 - 1.5 n^3 + O(n^2).

In "Visualizing the Sum-Product Conjecture", the bound a(n) <= (1/4)*(24 - 40*n + 27*n^2 - 8*n^3 + n^4) = 0.25 n^4 -2 n^3 +O(n^2) is proved.

From "Visualizing the Sum-Product Conjecture": 153 <= a(7) <= 157, 305 <= a(8) <= 358, 543 <= a(9) <= 645, 914 <= a(10) <= 1081, 1444 <= a(11) <= 1711, 2185 <= a(12) <= 2586, 3198 <= a(13) <= 3763, 4520 <= a(14) <= 5305, 6233 <= a(15) <= 7281, 8400 <= a(16) <= 9766. - Kevin O'Bryant, Mar 10 2025

Examples

			For n=3, the a(3) = 3 pairs are (5,6), (6,6), (6,5).
For n=4, the a(4) = 10 pairs are (u,10), (u,9), (10,8), (10,7), where 7 <= u <= 10.
		

Links

• Kevin O'Bryant, Visualizing the sum-product conjecture, arXiv:2411.08139 [math.NT], 2024.