OFFSET
1,1
COMMENTS
The successive values of q are 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, ... (see A246655).
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
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
EXAMPLE
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).
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
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
STATUS
approved