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%I #16 Apr 05 2022 00:11:39
%S 3,7,13,21,31,57,73,91,133,183,273,307,381,553,651,757,871,993,1057,
%T 1407,1723,1893,2257,2451,2863,3541,3783,4161,4557,5113,5403,6321,
%U 6643,6973,8011,9507,10303,10713,11557,11991,12883,14763,15751
%N 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).
%C For details on these difference sets see A333852, with references, and a W. Lang link.
%C 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.
%C From _Ed Pegg Jr_, May 16 2019: (Start)
%C (n^2+n+1,n+1) difference sets exist when n is a prime power.
%C (7,3), (1,2,4)
%C (13,4), (0,1,3,9)
%C (21,5), (3,6,7,12,14)
%C (31,6), (1,5,11,24,25,27)
%C (57,8), (0,1,6,15,22,26,45,55)
%C (73,9), (0,1,12,20,26,30,33,35,57)
%C (91,10), (0,2,6,7,18,21,31,54,63,71)
%C (133,12), (1,10,11,13,27,31,68,75,83,110,115,121)
%C (183,14), (1,13,20,21,23,44,61,72,77,86,90,116,122,169) (End)
%C Is a(n) = A138077(n-1)? - _R. J. Mathar_, Sep 11 2020
%H Dan Gordon, <a href="https://dmgordon.org/diffset">Difference Sets</a>
%F a(n) = m(n)^2 + m(n) + 1 , with m(n) = A000961(n), for n >= 1.
%e 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.
%Y Cf. A000961, A333852, A335866, A138077.
%K nonn,easy
%O 1,1
%A _Wolfdieter Lang_, Jul 26 2020
%E Comments about difference sets moved from A138077 to here by _Max Alekseyev_, Apr 05 2022
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