OFFSET
4,1
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.
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
EXAMPLE
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.
CROSSREFS
KEYWORD
sign
AUTHOR
Martin Becker, May 03 2025
STATUS
approved