A396871 a(n) = 4*5^(2*n+1) - floor(2*5^(n+1/2))^2.

4, 16, 179, 19, 475, 11875, 17371, 434275, 375316, 9382900, 59879696, 186796331, 302588011, 7564700275, 25342996816, 87659886836, 826709586979, 195925915636, 4898147890900, 122453697272500, 502365711956491, 4029220399392211, 15431285989604579, 65910059758111796

Offset

0,1

Comments

a(n) is the negated discriminants of the finite fields GF(3^(2*n+1)). Discriminants of finite fields defined in the thesis of Zaitsev and the Alekseenko et al paper to study algebraic curves over finite fields.

a(n) -> oo as n -> oo since each number can only occur finitely times: for each d, the equation x^2 + d = 4*5^n can be partitioned into three elliptic curves x^2 + d = 4*Y^3, x^2 + d = 4*5*Y^3, and x^2 + d = 4*5^2*Y^3, each having only finitely many integer solutions.

Links

Jianing Song, Table of n, a(n) for n = 0..500

E. Alekseenko, S. Aleshnikov, N. Markin and A. Zaytsev, Optimal Curves of Genus 3 over Finite Fields with Discriminant -19, arXiv:0902.1901 [math.AG], 2009-2011. See Definition 1.1.

A.I. Zaitsev, Optimality properties of curves over finite fields, University of Armsterdam. See Section 3.1.

Mathematica

A396871[n_] := 4*5^(2*n + 1) - Floor[2*5^(n + 1/2)]^2;
Array[A396871, 25, 0] (* Paolo Xausa, Jun 09 2026 *)

Prog

(PARI) a(n, p=5) = my(r); sqrtint(4*p^(2*n+1), &r); r

Crossrefs

Cf. A396869.

Cf. discriminants of finite fields whose order is an odd power of p: A201125 (p=2), A396870 (p=3), this sequence (p=5), A396872 (p=7).

Sequence in context: A033911 A130691 A308239 * A382930 A012916 A012921

Adjacent sequences: A396868 A396869 A396870 * A396872 A396873 A396874

Keyword

nonn

Author

Jianing Song, Jun 08 2026

Status

approved