A365877 a(n) is the number of quadratic equations u*x^2 + v*x + w = 0 with distinct solution sets L != {} and integer coefficients u, v, w, where n >= abs(u) + abs(v) + abs(w) and the sum of the solutions equals the product of the solutions.

1, 1, 2, 3, 5, 6, 9, 11, 15, 17, 23, 25, 32, 35, 40, 44, 53, 56, 66, 71, 78, 83, 95, 100, 111, 117, 127, 134, 150, 154, 171, 180, 191, 199, 213, 220, 240, 250, 263, 272, 294, 301, 324, 335, 348, 360, 386, 395, 419, 430, 448, 461, 490, 500, 522, 536, 556, 571, 603

Offset

1,3

Comments

According to Vieta's formulas, x_1 + x_2 = -v/u and x_1*x_2 = w/u. So x_1 + x_2 = x_1*x_2 when v = -w. Furthermore, the discriminant must not be negative, i.e., v^2 - 4*u*w = v^2 + 4*u*v >= 0.

Links

Table of n, a(n) for n=1..59.

Wikipedia, Quadratic equation.

Wikipedia, Vieta's formulas.

Formula

a(n) = Sum_{k=1..n} A365876(k).

a(n) = A341123(n) for 1 <= n <= 13.

Example

For n = 11 the a(11) = 23 equations are given by (u, v, w) = (1, 0, 0), (1, 1, -1), (2, 1, -1), (1, 2, -2), (3, 1, -1), (4, 1, -1), (5, 1, -1), (3, 2, -2), (1, 3, -3), (6, 1, -1), (2, 3, -3), (7, 1, -1), (5, 2, -2), (1, 4, -4), (-1, 4, -4), (8, 1, -1), (4, 3, -3), (9, 1, -1), (7, 2, -2), (5, 3, -3), (3, 4, -4), (1, 5, -5), (-1, 5, -5).
Equations multiplied by -1 do not have a different solution set; for example, (- 1, -1, 1) has the same solution set as (1, 1, -1).
Equations with GCD(u, v, w) != 1 are excluded, because their solution set are assigned to equations with lower n. For example, (2, 0, 0) is not included here, because its solution set is already assigned to (1, 0, 0).
Equations with a double solution are considered to have two equal solutions. For example, (-1, 4, -4) has the two solutions x_1 = x_2 = 2.

Maple

A365876:= proc(n) local u, v, a, min; u := n; v := 0; a := 0; min := true; while min = true do if u <> 0 and gcd(u, v) = 1 then a := a + 1; end if; u := u - 2; v:=(n-abs(u))/2; if u < -1/9*n then min := false; end if; end do; return a; end proc;
A365877:= proc(n) local s; option remember; if n = 1 then A365876(1); else procname(n - 1) + A365876(n); end if; end proc; seq(A365877(n), n = 1 .. 59);

Prog

(Python)
from math import gcd
def A365877(n):
    if n == 1: return 1
    c = 1
    for m in range(2, n+1):
        for v in range(1, m+1>>1):
            u = m-(v<<1)
            if gcd(u, v)==1:
                v2, u2 = v*v, v*(u<<2)
                if v2+u2 >= 0:
                    c +=1
                if v2-u2 >= 0:
                    c +=1
    return c # Chai Wah Wu, Oct 05 2023

Crossrefs

Cf. A364384, A364385, A341123, A365892.

Partial sums of A365876.

Sequence in context: A104738 A319469 A341123 * A325863 A341868 A342497

Adjacent sequences: A365874 A365875 A365876 * A365878 A365879 A365880

Keyword

nonn

Author

Felix Huber, Sep 22 2023

Status

approved