A213272 Costas arrays such that the terms in each row of the difference table are unique modulo n.

1, 2, 0, 8, 0, 12, 0, 0, 0, 40, 0, 48, 0, 0, 0, 128, 0, 108, 0, 0, 0, 220, 0, 0, 0, 0, 0, 336, 0

Offset: 1

Author

Joerg Arndt, Jun 08 2012

Keywords

• nonn
• hard
• more

Comments

Permutations of n elements such that each row in the difference table consists of pairwise distinct elements, even when taken modulo n (see example).

For n<=29 the nonzero terms a(n) appear for n in A006093 (primes minus 1) and a(n)=A002618(n) (n*phi(n)); omitting the zeros we obtain A104039 (number of primitive roots modulo (p(n))^2, where p(n) is n-th prime).

A002618(n) divides a(n) for all n, since (treating elements as integers modulo n) adding or subtracting a constant from each element or multiplying each element by an integer coprime to n preserves distinctness of all values modulo n. - Charlie Neder, May 26 2019

Examples

			The permutation (10, 9, 2, 8, 6, 1, 3, 7, 4, 5) corresponds to a Costas array:
  10  9  2  8  6  1  3  7  4  5  (Permutation: p(1), p(2), p(3), ..., p(n) )
  -1 -7  6 -2 -5  2  4 -3  1     (step-1 differences: p(2)-p(1), p(3)-p(2), ... )
  -8 -1  4 -7 -3  6  1 -2        (step-2 differences: p(3)-p(1), p(4)-p(2), ... )
  -2 -3 -1 -5  1  3  2           (step-3 differences: p(4)-p(1), p(5)-p(2), ... )
  -4 -8  1 -1 -2  4              ( etc. )
  -9 -6  5 -4 -1
  -7 -2  2 -3
  -3 -5  3
  -6 -4
  -5
The values in each row are unique also modulo n=10:
  10 9 2 8 6 1 3 7 4 5  (Permutation: p(1), p(2), p(3), ..., p(n) )
   9 3 6 8 5 2 4 7 1    (step-1 differences: p(2)-p(1), p(3)-p(2), ... )
   2 9 4 3 7 6 1 8      (step-2 differences: p(3)-p(1), p(4)-p(2), ... )
   8 7 9 5 1 3 2        (step-3 differences: p(4)-p(1), p(5)-p(2), ... )
   6 2 1 9 8 4          ( etc. )
   1 4 5 6 9
   3 8 2 7
   7 5 3
   4 6
   5
		

Links

• Scott Rickard, costasarrays.org (information and papers about Costas arrays). [broken link?]
• Wikipedia, Costas array.

Crossrefs

Cf. A008404 (Costas arrays), A213270 (Costas arrays that are involutions), A213271 (Costas arrays that are derangements), A213338 (Costas arrays that are cyclic), A213339 (Costas arrays that are connected).