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%I M0475 #31 Aug 27 2023 10:35:41
%S 2,3,4,5,6,7,8,9,10,12,13,14,16,18,19,20,21,22,25,27,28,30,32,34,37,
%T 38,40,42,44,45,48,50,51,54,58,61,62,64,65,67,72,74,75,75,77,80,81,87,
%U 88,91,94,96,98,100,103,104,109,110,113,114,120,126,129,130,132,135,136,137,141
%N Values k arising from a construction of Hirschfeld of k-arcs on elliptic curves over GF(q), where q = A246655(n) is the n-th prime power > 1.
%C Let E be an elliptic curve over GF(q). A k-arc on E is a set of k points in E(GF(q)) such that no three are collinear (in the projective plane over GF(q)). Hirschfeld showed that if the number #E(GF(q)) of GF(q)-rational points on E is even, then there exists a k-arc on E for k = #E(GF(q))/2. Here, a(n) denotes the largest possible k arising from this construction, hence a(n) = floor(A005523(n)/2). Note that a(n) is not necessarily the maximal k such that there exists a k-arc on an elliptic curve over GF(q); e.g. the elliptic curve y^2 = x^3 + x + 1 over GF(5) contains a 6-arc consisting of the points {(0,1), (3,1), (4,2), (4,3), (0,4), (3,4)}. - _Robin Visser_, Aug 26 2023
%D 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. See M_q(1) on page 51.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Robin Visser, <a href="/A005524/b005524.txt">Table of n, a(n) for n = 1..10000</a>
%H J. W. P. Hirschfeld, G. Korchmáros, and F. Torres, <a href="https://doi.org/10.1515/9781400847419">Algebraic curves over a finite field</a>, Princeton Ser. Appl. Math., Princeton University Press, Princeton, NJ, 2008.
%H Mathematica Information Center, <a href="http://library.wolfram.com/infocenter/MathSource/5175/">Item 5175</a>, for full code.
%H Ed Pegg Jr, <a href="https://web.archive.org/web/20130516043147/http://www.maa.org/editorial/mathgames/mathgames_04_12_04.html">Integer Complexity</a>.
%F a(n) = floor(A005523(n)/2) [Hirschfeld]. - _Robin Visser_, Aug 26 2023
%e For n = 4, the elliptic curve E : y^2 = x^3 + 3*x over GF(5) has 10 rational points. As this is the maximal number of rational points an elliptic curve over GF(5) can have, this implies a(4) = 10/2 = 5. - _Robin Visser_, Aug 26 2023
%o (Sage)
%o for q in range(1, 1000):
%o if Integer(q).is_prime_power():
%o p = Integer(q).prime_factors()[0]
%o if (floor(2*sqrt(q))%p != 0) or (Integer(q).is_square()) or (q==p):
%o print(floor((q + 1 + floor(2*sqrt(q)))/2))
%o else:
%o print(floor((q + floor(2*sqrt(q)))/2)) # _Robin Visser_, Aug 26 2023
%Y Cf. A000961 (values of q), A005523, A365216.
%K nonn
%O 1,1
%A _N. J. A. Sloane_.
%E New name and more terms from _Robin Visser_, Aug 26 2023
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