A365216 Maximal k such that there exists a k-arc on the projective plane over GF(q), where q = A246655(n) is the n-th prime power > 1.

4, 4, 6, 6, 8, 10, 10, 12, 14, 18, 18, 20, 24, 26, 28, 30, 32, 34, 38, 42, 44, 48, 50, 54, 60, 62, 66, 68, 72, 74, 80, 82, 84, 90, 98, 102, 104, 108, 110, 114, 122, 126, 128, 130, 132, 138, 140, 150, 152, 158, 164, 168, 170, 174, 180, 182, 192, 194, 198, 200, 212, 224, 228, 230, 234

Offset: 1

Author

Robin Visser, Aug 26 2023

Keywords

• nonn

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.

Examples

			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.
		

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.

Crossrefs

Cf. A000509, A005524.

Programs

Mathematica

Map[#+2-Mod[#,2]&,Select[Range[200],PrimePowerQ]] (* Paolo Xausa, Oct 23 2023 *)

Sage

for q in range(2, 1000):
    if Integer(q).is_prime_power(): print(q + 2 - (q%2))

Formula

a(n) = q + 1 if q is odd, otherwise a(n) = q + 2, where q = A246655(n).