cp's OEIS Frontend

A364690 Prime powers q such that there does not exist an elliptic curve E over GF(q) with cardinality q + 1 + floor(2*sqrt(q)).

%I A364690 #23 Feb 04 2025 01:30:17
%S A364690 128,2048,2187,16807,32768,131072,524288,1953125,2097152,8388608,
%T A364690 14348907,48828125,134217728,536870912,30517578125,549755813888,
%U A364690 847288609443,2199023255552,19073486328125,140737488355328,562949953421312,36028797018963968,144115188075855872,450283905890997363
%N A364690 Prime powers q such that there does not exist an elliptic curve E over GF(q) with cardinality q + 1 + floor(2*sqrt(q)).
%C A364690 By Hasse's theorem, every elliptic curve E over GF(q) has cardinality at most q + 1 + floor(2*sqrt(q)).  Moreover, for every prime power q, there exists an elliptic curve E over GF(q) with cardinality at least q + floor(2*sqrt(q)).  Thus these are the prime powers q for which A005523(n) = q + floor(2*sqrt(q)), where q = A246655(n).
%C A364690 By a theorem of Deuring and Waterhouse, these are exactly the prime powers q = p^k such that q is not prime, q is not a square, and p divides floor(2*sqrt(q)).
%H A364690 Katie Ahrens and Jon Grantham, <a href="/A364690/b364690.txt">Table of n, a(n) for n = 1..146</a>
%H A364690 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 A364690 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.
%e A364690 The first few values of the sequence (factorized) are 2^7, 2^11, 3^7, 7^5, 2^15, 2^17, 2^19, 5^9, 2^21, 2^23, 3^15, 5^11, 2^27, 2^29, ...
%o A364690 (Sage)
%o A364690 for q in range(1, 100000):
%o A364690     if Integer(q).is_prime_power():
%o A364690         p = Integer(q).prime_factors()[0]
%o A364690         if (floor(2*sqrt(q))%p == 0) and (not Integer(q).is_square()) and (q != p):
%o A364690             print(q)
%Y A364690 Cf. A005523, A246655.
%Y A364690 Subsequence of A246547.
%K A364690 nonn
%O A364690 1,1
%A A364690 _Robin Visser_, Aug 02 2023