A383574 Fourth column of A353077.

9, 14, 8, -1, 13, 7, 9, -1, 12, -1, 16, -1, -1, 7, 21, -1, 12, -1, -1, -1, 13, -1, 33, -1, 9, -1, 12, -1, 13, 7, -1, -1, -1, -1, 19, -1, -1, -1, 8, -1, 10, -1, -1, -1, 10, -1, 25, -1, -1, -1, 15, -1, -1, -1, -1, -1, 8, -1, 16, -1, -1, 7, -1, -1, 12, -1, -1

Offset: 4

Author

Martin Becker, May 03 2025

Keywords

• sign

Comments

Integers 0 to 6 are not in the sequence: For n > 5, the first three columns of A353077 are necessarily -1, -1, -1 or 0, 1, 3, and the fourth column is -1 or > 3, respectively. It is actually > 6 in the second case, as 4 - 3 = 1 - 0, 5 - 3 = 3 - 1, 6 - 3 = 3 - 0, respectively, would violate the distinctness of differences in perfect difference sets.

For n = 2^m + 1, m > 2, a(n) = 7, because 2 is a multiplier of such sets, therefore perfect difference sets containing 1, 2, 4, and 8 with translated sets containing 0, 1, 3, and 7 exist.

If n-1 is a prime power, a(n) != -1, as then there exist Singer type perfect difference sets.

If 4 <= n < 2*10^10 and n-1 is not a prime power, a(n) = -1. Cf. Gordon (2020).

Empirical observations further suggest that:

For n = 3^m + 1, m >= 1, a(n) = 9.

The most frequent positive value is 10.

11 is not in the sequence.

Examples

			For n = 4, there are 4 perfect difference sets containing 0 and 1: {0, 1, 3, 9}, {0, 1, 4, 6}, {0, 1, 5, 11}, and {0, 1, 8, 10}. The lexically earliest is {0, 1, 3, 9}. Its fourth element is 9, thus a(4) = 9.
There are no perfect difference sets with 7 elements. Thus a(7) = -1.
		

Links

• Martin Becker, Table of n, a(n) for n = 4..5000
• Daniel Gordon, The Prime Power Conjecture is true for n < 2,000,000, 1994.
• Daniel Gordon, On difference sets with small lambda, arXiv:2007.07292 [math.CO], 2020.
• Eric Weisstein's World of Mathematics, Perfect Difference Set
• Eric Weisstein's World of Mathematics, Prime Power Conjecture

Crossrefs

Fourth column of A353077.

Cf. A000961, A333852, A373514.