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A005523 a(n) = maximal number of rational points on an elliptic curve over GF(q), where q = A246655(n) is the n-th prime power > 1.

%I A005523 M3757 #51 Feb 16 2025 08:32:28
%S A005523 5,7,9,10,13,14,16,18,21,25,26,28,33,36,38,40,43,44,50,54,57,61,64,68,
%T A005523 75,77,81,84,88,91,97,100,102,108,117,122,124,128,130,135,144,148,150,
%U A005523 150,154,161,163,174,176,183,189,193,196,200,206,208,219,221,226,228,241,253,258,260
%N A005523 a(n) = maximal number of rational points on an elliptic curve over GF(q), where q = A246655(n) is the n-th prime power > 1.
%C A005523 The successive values of q are 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, ... (see A246655).
%D A005523 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 N_q(1) on page 51.
%D A005523 J.-P. Serre, Oeuvres, vol. 3, pp. 658-663 and 664-669.
%D A005523 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005523 Robin Visser, <a href="/A005523/b005523.txt">Table of n, a(n) for n = 1..10000</a>
%H A005523 Max Deuring, <a href="https://doi.org/10.1007/BF02940746">Die Typen der Multiplikatorenringe elliptischer Funktionenkörper</a>, Abh. Math. Sem. Hansischen Univ. 14 (1941), 197-272.
%H A005523 W. C. Waterhouse, <a href="https://doi.org/10.24033/asens.1183">Abelian varieties over finite fields</a>, Ann Sci. E.N.S., (4) 2 (1969), 521-560.
%H A005523 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RationalPoint.html">Rational Point.</a>
%H A005523 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hasse%27s_theorem_on_elliptic_curves">Hasse's theorem on elliptic curves</a>
%F A005523 a(n) <= q + 1 + 2*sqrt(q) where q = A246655(n) [Hasse theorem]. - _Sean A. Irvine_, Jun 26 2020
%F A005523 a(n) = q + 1 + floor(2*sqrt(q)) if p does not divide floor(2*sqrt(q)), q is a square, or q = p.  Otherwise a(n) = q + floor(2*sqrt(q)) where q = A246655(n) [Waterhouse 1969]. - _Robin Visser_, Aug 02 2023
%e A005523 a(1) = 5 because 5 is the maximal number of rational points on an elliptic curve over GF(2),
%e A005523 a(2) = 7 because 7 is the maximal number of rational points on an elliptic curve over GF(3),
%e A005523 a(3) = 9 because 9 is the maximal number of rational points on an elliptic curve over GF(4).
%o A005523 (Sage)
%o A005523 for q in range(1, 1000):
%o A005523     if Integer(q).is_prime_power():
%o A005523         p = Integer(q).prime_factors()[0]
%o A005523         if (floor(2*sqrt(q))%p != 0) or (Integer(q).is_square()) or (q==p):
%o A005523             print(q + 1 + floor(2*sqrt(q)))
%o A005523         else:
%o A005523             print(q + floor(2*sqrt(q)))  # _Robin Visser_, Aug 02 2023
%Y A005523 Cf. A000961, A246655.
%K A005523 nonn,easy
%O A005523 1,1
%A A005523 _N. J. A. Sloane_
%E A005523 Reworded definition and changed offset so as to clarify the indexing. - _N. J. A. Sloane_, Jan 08 2017
%E A005523 More terms from _Robin Visser_, Aug 02 2023