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A353077 Triangle read by rows, where the n-th row consists of the lexicographically earliest solution for n integers in 0..p-1 whose n*(n-1) differences are congruent to 1..p-1 (mod p), where p=n*(n-1)+1. If no solution exists, the n-th row consists of n -1's.

%I A353077 #53 May 28 2025 09:18:00
%S A353077 0,0,1,0,1,3,0,1,3,9,0,1,4,14,16,0,1,3,8,12,18,-1,-1,-1,-1,-1,-1,-1,0,
%T A353077 1,3,13,32,36,43,52,0,1,3,7,15,31,36,54,63,0,1,3,9,27,49,56,61,77,81,
%U A353077 -1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,3,12,20,34,38,81,88,94,104,109
%N A353077 Triangle read by rows, where the n-th row consists of the lexicographically earliest solution for n integers in 0..p-1 whose n*(n-1) differences are congruent to 1..p-1 (mod p), where p=n*(n-1)+1. If no solution exists, the n-th row consists of n -1's.
%C A353077 Comment from _Martin Becker_, May 18 2025: (Start)
%C A353077 Rows k with k-1 not a prime power are precisely the rows with -1 values for k <= 2*10^10. Cf. the Gordon (2020) link.
%C A353077 In the b-file, values a(n) > 3 from row 17 onwards depend on the conjecture that all perfect difference sets are Singer type, and were obtained through computer enumeration of Singer type sets.
%C A353077 (End)
%H A353077 Martin Becker, <a href="/A353077/b353077.txt">Rows n = 1..200 of triangle, flattened</a>.
%H A353077 Leonard E. Dickson, <a href="https://doi.org/10.2307/2968498">Problem 142</a>, The American Mathematical Monthly, Vol. 14, No. 5 (May, 1907), pp. 107-108.
%H A353077 Daniel Gordon, <a href="https://arxiv.org/abs/2007.07292">On difference sets with small lambda</a>, arXiv:2007.07292 [math.CO], 2020.
%H A353077 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PerfectDifferenceSet.html">Perfect Difference Set</a>
%e A353077    n   row
%e A353077    1 [0];
%e A353077    2 [0,1];
%e A353077    3 [0,1,3];
%e A353077    4 [0,1,3,9];
%e A353077    5 [0,1,4,14,16];
%e A353077    6 [0,1,3,8,12,18];
%e A353077    7 no solution exists;
%e A353077    8 [0,1,3,13,32,36,43,52];
%e A353077    9 [0,1,3,7,15,31,36,54,63];
%e A353077   10 [0,1,3,9,27,49,56,61,77,81];
%e A353077   11 no solution exists;
%e A353077   12 [0,1,3,12,20,34,38,81,88,94,104,109];
%e A353077   13 no solution exists;
%e A353077   14 [0,1,3,16,23,28,42,76,82,86,119,137,154,175];
%e A353077   15 no solution exists;
%e A353077   16 no solution exists.
%o A353077 (PARI) isok(n, v) = my(p=n*(n-1)+1); setbinop((x,y)->lift(Mod(x-y, p)), v, v) == [0..p-1];
%o A353077 row(n) = forsubset([n^2-n+1, n], s, my(ds = apply(x->x-1, Vec(s))); if (isok(n, ds), return(ds)););
%Y A353077 Cf. A002061, A058241, A333852, A351690, A383574.
%K A353077 sign,look,tabl
%O A353077 1,6
%A A353077 _Michel Marcus_, Apr 22 2022
%E A353077 Name and data corrected for "lexicographically earliest solution" by _Michel Marcus_, May 09 2022
%E A353077 Adjusted to a regular triangle, and rows 1, 2, 7, and 10-12 inserted by _Pontus von Brömssen_, May 09 2022