A364690 -id:A364690 - cp's OEIS frontend

2, 3, 5, 7, 8, 13, 17, 31, 32, 37, 43, 73, 101, 128, 157, 197, 211, 241, 257, 307, 343, 401, 421, 463, 577, 601, 677, 757, 1123, 1297, 1483, 1601, 1723, 2048, 2187, 2551, 2917, 2971, 3137, 3307, 3541, 3907, 4357, 4423, 4831, 5113, 5477, 5701, 6007, 6163, 6481, 7057, 8011, 8101, 8191

Offset: 1

N. J. A. Sloane, Oct 21 2015

nonn

See Hirschfeld, pp. 49-50 for precise definition.

By a theorem of Hasse-Weil and Serre, every (absolutely irreducible, smooth) genus 2 curve over GF(q) has cardinality at most q + 1 + 2*floor(2*sqrt(q)). This sequence consists exactly of the prime powers q (excluding 4 and 9) for which there does not exist any genus 2 curve over GF(q) with cardinality equal to q + 1 + 2*floor(2*sqrt(q)). - Robin Visser, Aug 26 2023

J. W. P. Hirschfeld, Linear codes and algebraic codes, pp. 35-53 of F. C. Holroyd and R. J. Wilson, editors, Geometrical Combinatorics. Pitman, Boston, 1984.
J.-P. Serre, Oeuvres, vol. 3, pp. 658-663 and 664-669.

Robin Visser, Table of n, a(n) for n = 1..10000
Jean-Pierre Serre, Sur le nombre des points rationnels d'une courbe algébrique sur un corps fini, C. R. Acad. Sci. Paris Ser. I Math. 296 (1983), no. 9, 397-402.
Jean-Pierre Serre, Nombres de points des courbe algebriques sur F_q, Sémin. Théorie Nombres Bordeaux, 1982/83, No. 22; Oeuvres, vol. 3, pp. 664-669.

Cf. A005525, A364690.

Subsequence of A246655.

for q in range(1, 1000):
    if Integer(q).is_prime_power():
        p = Integer(q).prime_factors()[0]
        if (not Integer(q).is_square()):
            if ((floor(2*sqrt(q))%p == 0) or (q-1).is_square() or
                (4*q-3).is_square() or (4*q-7).is_square()): print(q) # Robin Visser, Aug 26 2023

More terms from Robin Visser, Aug 26 2023

cp's OEIS Frontend

A262587 "Special" prime powers in Serre's sense.

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