A364690 - OEIS

%I #23 Feb 04 2025 01:30:17

%S 128,2048,2187,16807,32768,131072,524288,1953125,2097152,8388608,

%T 14348907,48828125,134217728,536870912,30517578125,549755813888,

%U 847288609443,2199023255552,19073486328125,140737488355328,562949953421312,36028797018963968,144115188075855872,450283905890997363

%N 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 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 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 Katie Ahrens and Jon Grantham, <a href="/A364690/b364690.txt">Table of n, a(n) for n = 1..146</a>

%H 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 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 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 (Sage)

%o for q in range(1, 100000):

%o     if Integer(q).is_prime_power():

%o         p = Integer(q).prime_factors()[0]

%o         if (floor(2*sqrt(q))%p == 0) and (not Integer(q).is_square()) and (q != p):

%o             print(q)

%Y Cf. A005523, A246655.

%Y Subsequence of A246547.

%K nonn,changed

%O 1,1

%A _Robin Visser_, Aug 02 2023