OFFSET
1,1
COMMENTS
A k-arc is a set of k points in PG(2,q) (the projective plane over GF(q)) such that no three are collinear.
REFERENCES
J. W. P. Hirschfeld, Linear codes and algebraic curves, pp. 35-53 of F. C. Holroyd and R. J. Wilson, editors, Geometrical Combinatorics. Pitman, Boston, 1984.
LINKS
Paolo Xausa, Table of n, a(n) for n = 1..10000
J. W. P. Hirschfeld, G. Korchmáros, and F. Torres, Algebraic curves over a finite field, Princeton Ser. Appl. Math., Princeton University Press, Princeton, NJ, 2008.
FORMULA
a(n) = q + 1 if q is odd, otherwise a(n) = q + 2, where q = A246655(n).
EXAMPLE
For n = 1, the four points (0:0:1), (1:0:1), (0:1:1), (1:1:1) form a 4-arc in PG(2,2); the projective plane over GF(2). Moreover, any five points in PG(2,2) contain three points which are collinear, thus a(1) = 4.
For n = 4, the six points (0:0:1), (1:0:1), (0:1:1), (1:1:1), (3:2:1), (3:4:1) form a 6-arc in PG(2,5); the projective plane over GF(5). Moreover, any seven points in PG(2,5) contain three points which are collinear, thus a(4) = 6.
MATHEMATICA
Map[#+2-Mod[#, 2]&, Select[Range[200], PrimePowerQ]] (* Paolo Xausa, Oct 23 2023 *)
PROG
(Sage)
for q in range(2, 1000):
if Integer(q).is_prime_power(): print(q + 2 - (q%2))
CROSSREFS
KEYWORD
nonn
AUTHOR
Robin Visser, Aug 26 2023
STATUS
approved