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A213272
Costas arrays such that the terms in each row of the difference table are unique modulo n.
0
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
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
KEYWORD
nonn,hard,more
AUTHOR
Joerg Arndt, Jun 08 2012
STATUS
approved