

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.


2



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
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OFFSET

1,2


COMMENTS

For details on these simple difference sets see A333852, with references, and a W. Lang link.
The formula given below has been conjectured by Singer for n >= 2 on p. 383. See also the table on p. 384.


LINKS

Table of n, a(n) for n=1..59.
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

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 power of the prime dividing m(n), for n >= 2.


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}}.


CROSSREFS

Cf. A000010, A025474, A000961, A333852, A335865.
Sequence in context: A305260 A162982 A259707 * A247085 A320971 A342828
Adjacent sequences: A335863 A335864 A335865 * A335867 A335868 A335869


KEYWORD

nonn,easy


AUTHOR

Wolfdieter Lang, Jul 26 2020


STATUS

approved



