A095029 -id:A095029 - cp's OEIS frontend

A095025 Number of inequivalent cyclic difference sets with n elements.

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

A095047 An example of a (v,k,lambda)=(107,53,26) cyclic difference set.

Original entry on oeis.org

1, 3, 4, 9, 10, 11, 12, 13, 14, 16, 19, 23, 25, 27, 29, 30, 33, 34, 35, 36, 37, 39, 40, 41, 42, 44, 47, 48, 49, 52, 53, 56, 57, 61, 62, 64, 69, 75, 76, 79, 81, 83, 85, 86, 87, 89, 90, 92, 99, 100, 101, 102, 105

Offset: 1

Hugo Pfoertner, May 27 2004

See A095029. This is an example of a difference set that produces a large number of different representations.

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-A095046 (more examples of cyclic difference sets with k=5..20).

A335865 Moduli a(n) = v(n) for the simple difference sets of Singer type of order m(n) (v(n), m(n)+1, 1) in the additive group modulo v(n) = m(n)^2 + m(n) + 1, with m(n) = A000961(n).

Original entry on oeis.org

3, 7, 13, 21, 31, 57, 73, 91, 133, 183, 273, 307, 381, 553, 651, 757, 871, 993, 1057, 1407, 1723, 1893, 2257, 2451, 2863, 3541, 3783, 4161, 4557, 5113, 5403, 6321, 6643, 6973, 8011, 9507, 10303, 10713, 11557, 11991, 12883, 14763, 15751

Offset: 1

Wolfdieter Lang, Jul 26 2020

For details on these difference sets see A333852, with references, and a W. Lang link.
Because these simple difference sets of Singer type of order m = m(n) in the addive group (Z_{v(n)}, +) = RS(v(n)) = {0, 1, ..., v(n)-1} are also simple symmetric balanced incomplete block designs (BIBD), the number of blocks b(n) is also v(n) = a(n). This is the number of simple difference sets of each of the A335865(n) classes.
From Ed Pegg Jr, May 16 2019, edited by Hugo Pfoertner, May 13 2024: (Start)
(n^2+n+1,n+1) difference sets exist when n is a prime power.
(7,3), (1,2,4)
(13,4), (0,1,3,9)
(21,5), (3,6,7,12,14) (A095029)
(31,6), (1,5,11,24,25,27) (A095030)
(57,8), (0,1,6,15,22,26,45,55) (A095032)
(73,9), (0,1,12,20,26,30,33,35,57) (A095035)
(91,10), (0,2,6,7,18,21,31,54,63,71) (A095036)
(133,12), (1,10,11,13,27,31,68,75,83,110,115,121) (A095038)
(183,14), (1,13,20,21,23,44,61,72,77,86,90,116,122,169) (A095040) (End)
Is a(n) = A138077(n-1)? - R. J. Mathar, Sep 11 2020

			n = 2, m(2) = 2, a(2) = 2^2 + 2 + 1 = 7. The simple Singer difference set of order 2 is denoted by (7, 3, 1) (Fano plane). There are two classes (A335866(2) = 2) obtained from the representative difference sets {0, 1, 3} and {0, 1, 5} by element-wise addition of 1, 2, ..., 6 taken modulo 7. Each class consists of 7 simple difference sets.

Dan Gordon, Difference Sets

Cf. A000961, A333852, A335866, A138077.

a(n) = m(n)^2 + m(n) + 1 , with m(n) = A000961(n), for n >= 1.

Comments about difference sets moved from A138077 to here by Max Alekseyev, Apr 05 2022

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).

cp's OEIS Frontend

A095025 Number of inequivalent cyclic difference sets with n elements.

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A095047 An example of a (v,k,lambda)=(107,53,26) cyclic difference set.

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A335865 Moduli a(n) = v(n) for the simple difference sets of Singer type of order m(n) (v(n), m(n)+1, 1) in the additive group modulo v(n) = m(n)^2 + m(n) + 1, with m(n) = A000961(n).

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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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