A396872 a(n) = 4*7^(2*n+1) - floor(2*7^(n+1/2))^2.

3, 3, 147, 3576, 22812, 50616, 1235107, 8226931, 37066147, 107991483, 2302145628, 8174839347, 107602297939, 145628072796, 7135775567004, 48192192640248, 251191768371228, 2460676852185676, 17172107878050712, 117625880871152716, 697016326613157592, 4598330960283656403

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*7^n can be partitioned into three elliptic curves x^2 + d = 4*Y^3, x^2 + d = 4*7*Y^3, and x^2 + d = 4*7^2*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

A396872[n_] := 4*7^(2*n + 1) - Floor[2*7^(n + 1/2)]^2;
Array[A396872, 25, 0] (* Paolo Xausa, Jun 09 2026 *)

Prog

(PARI) a(n, p=7) = 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), A396871 (p=5), this sequence (p=7).

Sequence in context: A230646 A006845 A071536 * A094755 A152418 A273925

Adjacent sequences: A396869 A396870 A396871 * A396873 A396874 A396875

Keyword

nonn

Author

Jianing Song, Jun 08 2026

Status

approved