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

%I A262587 #19 Aug 28 2023 08:28:32
%S A262587 2,3,5,7,8,13,17,31,32,37,43,73,101,128,157,197,211,241,257,307,343,
%T A262587 401,421,463,577,601,677,757,1123,1297,1483,1601,1723,2048,2187,2551,
%U A262587 2917,2971,3137,3307,3541,3907,4357,4423,4831,5113,5477,5701,6007,6163,6481,7057,8011,8101,8191
%N A262587 "Special" prime powers in Serre's sense.
%C A262587 See Hirschfeld, pp. 49-50 for precise definition.
%C A262587 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
%D A262587 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.
%D A262587 J.-P. Serre, Oeuvres, vol. 3, pp. 658-663 and 664-669.
%H A262587 Robin Visser, <a href="/A262587/b262587.txt">Table of n, a(n) for n = 1..10000</a>
%H A262587 Jean-Pierre Serre, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k55351747/f35.item">Sur le nombre des points rationnels d'une courbe algébrique sur un corps fini</a>, C. R. Acad. Sci. Paris Ser. I Math. 296 (1983), no. 9, 397-402.
%H A262587 Jean-Pierre Serre, <a href="https://eudml.org/doc/182160">Nombres de points des courbe algebriques sur F_q</a>, Sémin. Théorie Nombres Bordeaux, 1982/83, No. 22; Oeuvres, vol. 3, pp. 664-669.
%o A262587 (Sage)
%o A262587 for q in range(1, 1000):
%o A262587     if Integer(q).is_prime_power():
%o A262587         p = Integer(q).prime_factors()[0]
%o A262587         if (not Integer(q).is_square()):
%o A262587             if ((floor(2*sqrt(q))%p == 0) or (q-1).is_square() or
%o A262587                 (4*q-3).is_square() or (4*q-7).is_square()): print(q) # _Robin Visser_, Aug 26 2023
%Y A262587 Cf. A005525, A364690.
%Y A262587 Subsequence of A246655.
%K A262587 nonn
%O A262587 1,1
%A A262587 _N. J. A. Sloane_, Oct 21 2015
%E A262587 More terms from _Robin Visser_, Aug 26 2023