A335866 - OEIS

%I #34 Jun 11 2024 09:40:07

%S 1,2,4,2,10,12,8,12,36,40,12,102,84,156,60,84,264,220,60,264,574,420,

%T 720,252,816,1180,768,144,840,1704,1200,1176,432,2196,2670,2112,3434,

%U 2380,3024,2280,3960,1296,1656,3612,672,5764,5184,3984,6120,4368,5512,4752,9352,3120,10034,9204,7176,9360,7128

%N 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.

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

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

%C This conjecture was later proved by Berman.

%H Martin Becker, <a href="/A335866/b335866.txt">Table of n, a(n) for n = 1..20000</a>

%H Gerald Berman, <a href="https://doi.org/10.1090/S0002-9947-1953-0054978-4">Finite projective plane geometries and difference sets</a>, Trans. AMS, 74 (1953) 492-499.

%H James Singer, <a href="https://doi.org/10.1090/S0002-9947-1938-1501951-4">A Theorem in Finite Projective Geometry and Some Applications to Number Theory</a>, Trans. AMS, 43 (1938) 377-385, Table on p. 384.

%F 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.

%e 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}}.

%o (PARI) print1(1); for(q=2, 193 , if(n=isprimepower(q), print1(", ", eulerphi(q^2+q+1)/(3*n)))) \\ _Martin Becker_, Jun 11 2024

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

%K nonn,easy

%O 1,2

%A _Wolfdieter Lang_, Jul 26 2020