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).

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

Author

Wolfdieter Lang, Jul 26 2020

Keywords

• nonn
• easy

Comments

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

Examples

			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.
		

Links

• Dan Gordon, Difference Sets

Crossrefs

Cf. A000961, A333852, A335866, A138077.

Formula

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

Extensions

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