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.

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, 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, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 0, 1, 3, 12, 20, 34, 38, 81, 88, 94, 104, 109

Offset

1,6

Comments

Comment from Martin Becker, May 18 2025: (Start)

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.

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.

(End)

Links

Martin Becker, Rows n = 1..200 of triangle, flattened.

Leonard E. Dickson, Problem 142, The American Mathematical Monthly, Vol. 14, No. 5 (May, 1907), pp. 107-108.

Daniel Gordon, On difference sets with small lambda, arXiv:2007.07292 [math.CO], 2020.

Eric Weisstein's World of Mathematics, Perfect Difference Set

Example

   n   row
   1 [0];
   2 [0,1];
   3 [0,1,3];
   4 [0,1,3,9];
   5 [0,1,4,14,16];
   6 [0,1,3,8,12,18];
   7 no solution exists;
   8 [0,1,3,13,32,36,43,52];
   9 [0,1,3,7,15,31,36,54,63];
  10 [0,1,3,9,27,49,56,61,77,81];
  11 no solution exists;
  12 [0,1,3,12,20,34,38,81,88,94,104,109];
  13 no solution exists;
  14 [0,1,3,16,23,28,42,76,82,86,119,137,154,175];
  15 no solution exists;
  16 no solution exists.

Prog

(PARI) isok(n, v) = my(p=n*(n-1)+1); setbinop((x, y)->lift(Mod(x-y, p)), v, v) == [0..p-1];
row(n) = forsubset([n^2-n+1, n], s, my(ds = apply(x->x-1, Vec(s))); if (isok(n, ds), return(ds)); );

Crossrefs

Cf. A002061, A058241, A333852, A351690, A383574.

Sequence in context: A325673 A083857 A388226 * A115142 A346203 A327126

Adjacent sequences: A353074 A353075 A353076 * A353078 A353079 A353080

Keyword

sign,look,tabl

Author

Michel Marcus, Apr 22 2022

Extensions

Name and data corrected for "lexicographically earliest solution" by Michel Marcus, May 09 2022

Adjusted to a regular triangle, and rows 1, 2, 7, and 10-12 inserted by Pontus von Brömssen, May 09 2022

Status

approved