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.

5, 7, 9, 10, 13, 14, 16, 18, 21, 25, 26, 28, 33, 36, 38, 40, 43, 44, 50, 54, 57, 61, 64, 68, 75, 77, 81, 84, 88, 91, 97, 100, 102, 108, 117, 122, 124, 128, 130, 135, 144, 148, 150, 150, 154, 161, 163, 174, 176, 183, 189, 193, 196, 200, 206, 208, 219, 221, 226, 228, 241, 253, 258, 260

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

The successive values of q are 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, ... (see A246655).

Examples

			a(1) = 5 because 5 is the maximal number of rational points on an elliptic curve over GF(2),
a(2) = 7 because 7 is the maximal number of rational points on an elliptic curve over GF(3),
a(3) = 9 because 9 is the maximal number of rational points on an elliptic curve over GF(4).
		

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. See N_q(1) on page 51.
• J.-P. Serre, Oeuvres, vol. 3, pp. 658-663 and 664-669.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Robin Visser, Table of n, a(n) for n = 1..10000
• Max Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkörper, Abh. Math. Sem. Hansischen Univ. 14 (1941), 197-272.
• W. C. Waterhouse, Abelian varieties over finite fields, Ann Sci. E.N.S., (4) 2 (1969), 521-560.
• Eric Weisstein's World of Mathematics, Rational Point.
• Wikipedia, Hasse's theorem on elliptic curves

Crossrefs

Cf. A000961, A246655.

Programs

Sage

for q in range(1, 1000):
    if Integer(q).is_prime_power():
        p = Integer(q).prime_factors()[0]
        if (floor(2*sqrt(q))%p != 0) or (Integer(q).is_square()) or (q==p):
            print(q + 1 + floor(2*sqrt(q)))
        else:
            print(q + floor(2*sqrt(q)))  # Robin Visser, Aug 02 2023

Formula

a(n) <= q + 1 + 2*sqrt(q) where q = A246655(n) [Hasse theorem]. - Sean A. Irvine, Jun 26 2020

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

Extensions

Reworded definition and changed offset so as to clarify the indexing. - N. J. A. Sloane, Jan 08 2017

More terms from Robin Visser, Aug 02 2023