This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A383574 #55 May 24 2025 10:36:05
%S A383574 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,
%T A383574 -1,12,-1,13,7,-1,-1,-1,-1,19,-1,-1,-1,8,-1,10,-1,-1,-1,10,-1,25,-1,
%U A383574 -1,-1,15,-1,-1,-1,-1,-1,8,-1,16,-1,-1,7,-1,-1,12,-1,-1
%N A383574 Fourth column of A353077.
%C A383574 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.
%C A383574 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.
%C A383574 If n-1 is a prime power, a(n) != -1, as then there exist Singer type perfect difference sets.
%C A383574 If 4 <= n < 2*10^10 and n-1 is not a prime power, a(n) = -1. Cf. Gordon (2020).
%C A383574 Empirical observations further suggest that:
%C A383574 For n = 3^m + 1, m >= 1, a(n) = 9.
%C A383574 The most frequent positive value is 10.
%C A383574 11 is not in the sequence.
%H A383574 Martin Becker, <a href="/A383574/b383574.txt">Table of n, a(n) for n = 4..5000</a>
%H A383574 Daniel Gordon, <a href="https://dmgordon.org/papers/ppc.pdf">The Prime Power Conjecture is true for n < 2,000,000</a>, 1994.
%H A383574 Daniel Gordon, <a href="https://arxiv.org/abs/2007.07292">On difference sets with small lambda</a>, arXiv:2007.07292 [math.CO], 2020.
%H A383574 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PerfectDifferenceSet.html">Perfect Difference Set</a>
%H A383574 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimePowerConjecture.html">Prime Power Conjecture</a>
%e A383574 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.
%e A383574 There are no perfect difference sets with 7 elements. Thus a(7) = -1.
%Y A383574 Fourth column of A353077.
%Y A383574 Cf. A000961, A333852, A373514.
%K A383574 sign
%O A383574 4,1
%A A383574 _Martin Becker_, May 03 2025