A095047 -id:A095047 - cp's OEIS frontend

A095029 An example of a (v,k,lambda)=(21,5,1) cyclic difference set.

3, 6, 7, 12, 14

Offset: 1

Hugo Pfoertner, May 27 2004

A (v,k,lambda) cyclic difference set is a subset D={d_1,d_2,...,d_k} of the integers modulo v such that {1,2,...,v-1} can each be represented as a difference (d_i-d_j) modulo v in exactly lambda different ways. Difference sets with lambda=1 (planar difference sets) have order n=k-1. The Prime Power Conjecture states that all Abelian planar difference sets have order n a prime power. It is known that no cyclic planar difference sets of nonprime power order n exist with n < 2*10^9 (see Baumert, Gordon link)

			Representation of {1,...,20}: 1=7-6, 2=14-12, 3=6-3, 4=7-3, 5=12-7, 6=12-6, 7=14-7, 8=14-6, 9=12-3, 10=21+3-14, 11=14-3, 12=21+3-12, 13=21+6-14, 14=21+7-14, 15=21+6-12, 16=21+7-12, 17=21+3-7, 18=21+3-6, 19=21+12-14, 20=21+6-7. - _Hugo Pfoertner_, Aug 13 2011

Leonard D. Baumert and Daniel M. Gordon, On the existence of cyclic difference sets with small parameters, arXiv:math/0304502 [math.CO], 30 Apr 2003.
Dan Gordon, List of Cyclic Difference Sets, (2003).
Dan Gordon, Difference Sets, searchable database.

Cf. A095025 (number of cyclic difference sets with n elements), A095029-A095047 (more examples of cyclic difference set with k=5..20), A000961 (prime powers).

A095025 Number of inequivalent cyclic difference sets with n elements.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 0, 2, 1, 0, 1, 2, 0, 1, 1, 1, 1, 0, 2, 1, 1, 3, 1, 3, 0, 1, 0, 0, 1, 1, 4, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 6, 0, 2, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0

Offset: 3

Hugo Pfoertner, May 27 2004

nonn

A (v,k,lambda) cyclic difference set is a subset D = {d_1, d_2, ..., d_k} of the integers modulo v such that {1, 2, ..., v-1} can each be represented as a difference (d_i-d_j) modulo v in exactly lambda different ways.

If D is a cyclic difference set, then D+a and u*D are again cyclic difference sets, for any a and any invertible u, cf. examples. Therefore this sequence counts only the equivalence classes of sets modulo such transformations. - M. F. Hasler, Jul 30 2021

			a(3) = 1 corresponds to the (7,3,1) set D = {1, 2, 4}: Each of {1, ..., 6} (mod 7) has exactly 1 representation as difference of two elements in D: 1 = 2 - 1; 2 = 4 - 2; 3 = 4 - 1; 4 == 1 - 4 (mod 7); 5 == 2 - 4 (mod 7); 6 == 1 - 2 (mod 7). The "shifted" sets {2, 3, 5}, {3, 4, 6}, {0, 4, 5}, {1, 5, 6}, {0, 2, 6}, {0, 1, 3} and -D == {3, 5, 6} == 3*D = -2*D and shifted variants of this set automatically also yield all elements of {1, ..., 6} (mod 7) exactly once as difference of two elements, but these "equivalent" variants are not counted separately.
a(4) = 1 corresponds to the (13,4,1) set D' = {0, 1, 3, 9}: again, each of {1, ..., 12} have exactly one representation as x - y (mod 13) with x, y in D'.
a(5) = 2 because there are two cyclic difference sets of length 5: The (v,k,lambda)=(11,5,2) set A095028 = {1, 3, 4, 5, 9} and the (21,5,1) set A095029 = {3, 6, 7, 12, 14}.

Ralf Goertz, Coprime ordering of cyclic planar difference sets, Discrete Mathematics, Volume 309, Issue 16, 2009, Pages 5248-5252.
Dan Gordon, Search Form for Difference Sets
Dan Gordon, La Jolla Difference Set Repository [from Internet Archive Wayback Machine]
Dan Gordon, List of Cyclic Difference Sets [from Internet Archive Wayback Machine]
Dan Gordon and Len Baumert, Papers on Difference Sets

Cf. A095029 - A095047: examples of cyclic difference set with 5 <= k <= 20.

Second example corrected by an anonymous reader - N. J. A. Sloane, Jul 19 2021

Definition clarified by M. F. Hasler, Jul 30 2021

A095028 An example of a (v,k,lambda)=(11,5,2) cyclic difference set.

Original entry on oeis.org

1, 3, 4, 5, 9

Offset: 1

Hugo Pfoertner, Jun 02 2004

See A095029.

			Using the numbers {1 3 4 5 9}, every number in the range 1..10 can be written as a difference modulo 11 in two different ways:
1=4-3=5-4, 2=3-1=5-3, 3=4-1=(1-9) mod 11, 4=5-1=9-5, 5=9-4=(3-9) mod 11,
6=9-2=(4-9) mod 11, 7=(1-5) mod 11=(5-9) mod 11, 8=9-1=(1-4) mod 11,
9=(1-3) mod 11=(3-5) mod 11, 10=(3-4) mod 11=(4-5) mod 11.

Dan Gordon, List of Cyclic Difference Sets, (2003).

Cf. A095025 number of cyclic difference sets with n elements, A095029 .. A095047 more examples of cyclic difference set with k=5..53.

A095030 An example of a (v,k,lambda)=(31,6,1) cyclic difference set.

Original entry on oeis.org

1, 5, 11, 24, 25, 27

Offset: 1

Hugo Pfoertner, May 27 2004

See A095029.

Dan Gordon, List of Cyclic Difference Sets, (2003).
Dan Gordon, Difference Sets, searchable database.

Cf. A095025 (number of cyclic difference sets with n elements), A095029-A095047 (more examples of cyclic difference set with k=5..20), A000961 (prime powers).

A095032 An example of a (v,k,lambda)=(57,8,1) cyclic difference set.

Original entry on oeis.org

0, 1, 6, 15, 22, 26, 45, 55

Offset: 1

Hugo Pfoertner, May 27 2004

See A095029.

Dan Gordon, List of Cyclic Difference Sets, (2003).
Dan Gordon, Difference Sets, searchable database.

Cf. A095025 (number of cyclic difference sets with n elements), A095029-A095047 (more examples of cyclic difference sets with k=5..20), A000961 (prime powers).

A095035 An example of a (v,k,lambda)=(73,9,1) cyclic difference set.

Original entry on oeis.org

0, 1, 12, 20, 26, 30, 33, 35, 57

Offset: 1

Hugo Pfoertner, May 27 2004

See A095029.

Dan Gordon, List of Cyclic Difference Sets, (2003).
Dan Gordon, Difference Sets, searchable database.

Cf. A095025 (number of cyclic difference sets with n elements), A095029-A095047 (more examples of cyclic difference sets with k=5..20), A000961 (prime powers).

A095036 An example of a (v,k,lambda)=(91,10,1) cyclic difference set.

Original entry on oeis.org

0, 2, 6, 7, 18, 21, 31, 54, 63, 71

Offset: 1

Hugo Pfoertner, May 27 2004

See A095029.

Dan Gordon, List of Cyclic Difference Sets, (2003).
Dan Gordon, Difference Sets, searchable database.

Cf. A095025 (number of cyclic difference sets with n elements), A095029-A095047 (more examples of cyclic difference sets with k=5..20), A000961 (prime powers).

A095038 An example of a (v,k,lambda)=(133,12,1) cyclic difference set.

Original entry on oeis.org

1, 10, 11, 13, 27, 31, 68, 75, 83, 110, 115, 121

Offset: 1

Hugo Pfoertner, May 27 2004

See A095029.

Dan Gordon, List of Cyclic Difference Sets, (2003).
Dan Gordon, Difference Sets, searchable database.

Cf. A095025 (number of cyclic difference sets with n elements), A095029-A095047 (more examples of cyclic difference sets with k=5..20), A000961 (prime powers).

A095040 An example of a (v,k,lambda)=(183,14,1) cyclic difference set.

Original entry on oeis.org

1, 13, 20, 21, 23, 44, 61, 72, 77, 86, 90, 116, 122, 169

Offset: 1

Hugo Pfoertner, May 27 2004

See A095029.

Dan Gordon, List of Cyclic Difference Sets, (2003).
Dan Gordon, Difference Sets, searchable database.

Cf. A095025 (number of cyclic difference sets with n elements), A095029-A095047 (more examples of cyclic difference sets with k=5..20), A000961 (prime powers).

A095041 One of two (v,k,lambda)=(31,15,7) cyclic difference sets. The other one is A095042.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 15, 16, 17, 23, 24, 27, 29, 30

Offset: 1

Hugo Pfoertner, May 27 2004

See A095029.

Dan Gordon, List of Cyclic Difference Sets, (2003).
Dan Gordon, Difference Sets, searchable database.

Cf. A095025 (number of cyclic difference sets with n elements), A095029-A095047 (more examples of cyclic difference sets with k=5..20).

cp's OEIS Frontend

A095029 An example of a (v,k,lambda)=(21,5,1) cyclic difference set.

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A095025 Number of inequivalent cyclic difference sets with n elements.

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A095028 An example of a (v,k,lambda)=(11,5,2) cyclic difference set.

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A095030 An example of a (v,k,lambda)=(31,6,1) cyclic difference set.

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A095032 An example of a (v,k,lambda)=(57,8,1) cyclic difference set.

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A095035 An example of a (v,k,lambda)=(73,9,1) cyclic difference set.

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A095036 An example of a (v,k,lambda)=(91,10,1) cyclic difference set.

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A095038 An example of a (v,k,lambda)=(133,12,1) cyclic difference set.

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A095040 An example of a (v,k,lambda)=(183,14,1) cyclic difference set.

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A095041 One of two (v,k,lambda)=(31,15,7) cyclic difference sets. The other one is A095042.

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