OFFSET
1,2
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.
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.
FORMULA
EXAMPLE
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}}.
PROG
(PARI) print1(1); for(q=2, 193 , if(n=isprimepower(q), print1(", ", eulerphi(q^2+q+1)/(3*n)))) \\ Martin Becker, Jun 11 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Jul 26 2020
STATUS
approved