A335866 Number of classes of simple difference sets of the Singer type (m^2 + m + 1, m + 1, 1) with m = m(n) = A000961(n), for n >= 1.

1, 2, 4, 2, 10, 12, 8, 12, 36, 40, 12, 102, 84, 156, 60, 84, 264, 220, 60, 264, 574, 420, 720, 252, 816, 1180, 768, 144, 840, 1704, 1200, 1176, 432, 2196, 2670, 2112, 3434, 2380, 3024, 2280, 3960, 1296, 1656, 3612, 672, 5764, 5184, 3984, 6120, 4368, 5512, 4752, 9352, 3120, 10034, 9204, 7176, 9360, 7128

Offset: 1

Author

Wolfdieter Lang, Jul 26 2020

Keywords

• nonn
• easy

Comments

For details on these simple difference sets see A333852, with references, and a W. Lang link.

The formula given below was conjectured by Singer for n >= 2 on p. 383. See also the table on p. 384.

This conjecture was later proved by Berman.

Examples

			n = 2, m(2) = 2 = 2^1, a(2) = phi(7)/(3*1) = 6/3 = 2. There are two classes of type (7,3,1) (Fano plane), with representatives {0, 1, 3} and {0, 1, 5}. The two equivalence classes (by elementwise addition of 1, 2, ..., 6 modulo 7) are Dev({0, 1, 3}) = {{0, 1, 3}, {0, 2, 6}, {0, 4, 5}, {1, 2, 4}, {1, 5, 6}, {2, 3, 5}, {3, 4, 6}, and Dev({0, 1, 5}) = {{0, 1, 5}, {0, 2, 3}, {0, 4, 6}, {1, 2, 6}, {1, 3, 4}, {2, 4, 5}, {3, 5, 6}}.
		

Links

• Martin Becker, Table of n, a(n) for n = 1..20000
• Gerald Berman, Finite projective plane geometries and difference sets, Trans. AMS, 74 (1953) 492-499.
• James Singer, A Theorem in Finite Projective Geometry and Some Applications to Number Theory, Trans. AMS, 43 (1938) 377-385, Table on p. 384.

Crossrefs

Cf. A000010, A025474, A000961, A333852, A335865.

Programs

PARI

print1(1); for(q=2, 193 , if(n=isprimepower(q), print1(", ", eulerphi(q^2+q+1)/(3*n)))) \\ Martin Becker, Jun 11 2024

Formula

a(1) = 1, and a(n) = phi(v(n))/(3*e(n)), with phi = A000010 (Euler's totient), v(n) = A335865(n) = m(n)^2 + m(n) + 1, with m(n) = A000961(n), and e(n) = A025474(n), the exponent of the prime power dividing m(n), for n >= 2.