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A335865
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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).
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5
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3, 7, 13, 21, 31, 57, 73, 91, 133, 183, 273, 307, 381, 553, 651, 757, 871, 993, 1057, 1407, 1723, 1893, 2257, 2451, 2863, 3541, 3783, 4161, 4557, 5113, 5403, 6321, 6643, 6973, 8011, 9507, 10303, 10713, 11557, 11991, 12883, 14763, 15751
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OFFSET
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1,1
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COMMENTS
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For details on these difference sets see A333852, with references, and a W. Lang link.
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.
(n^2+n+1,n+1) difference sets exist when n is a prime power.
(7,3), (1,2,4)
(13,4), (0,1,3,9)
(21,5), (3,6,7,12,14)
(31,6), (1,5,11,24,25,27)
(57,8), (0,1,6,15,22,26,45,55)
(73,9), (0,1,12,20,26,30,33,35,57)
(91,10), (0,2,6,7,18,21,31,54,63,71)
(133,12), (1,10,11,13,27,31,68,75,83,110,115,121)
(183,14), (1,13,20,21,23,44,61,72,77,86,90,116,122,169) (End)
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LINKS
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FORMULA
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a(n) = m(n)^2 + m(n) + 1 , with m(n) = A000961(n), for n >= 1.
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EXAMPLE
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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.
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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