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

1,2

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

Links

Table of n, a(n) for n=1..29.

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

Wikipedia, Costas array.

Example

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

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).

Sequence in context: A369869 A011250 A073830 * A053205 A167029 A094030

Adjacent sequences: A213269 A213270 A213271 * A213273 A213274 A213275

Keyword

nonn,hard,more

Author

Joerg Arndt, Jun 08 2012

Status

approved