A365876 a(n) is the number of quadratic equations u*x^2 + v*x + w = 0 with distinct solution sets L != {}, integer coefficients u, v, w and GCD(u, v, w) = 1, where n = abs(u) + abs(v) + abs(w) and the sum of the solutions equals the product of the solutions.

1, 0, 1, 1, 2, 1, 3, 2, 4, 2, 6, 2, 7, 3, 5, 4, 9, 3, 10, 5, 7, 5, 12, 5, 11, 6, 10, 7, 16, 4, 17, 9, 11, 8, 14, 7, 20, 10, 13, 9, 22, 7, 23, 11, 13, 12, 26, 9, 24, 11, 18, 13, 29, 10, 22, 14, 20, 15, 32, 9, 33, 16, 20, 18, 27, 11, 37, 18, 25, 13, 39, 13, 40, 20

Offset

1,5

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..74.

Wikipedia, Quadratic equation.

Wikipedia, Vieta's formulas.

Example

For n = 9 the a(9) = 4 equations are given by (u, v, w) = (7, 1, -1), (5, 2, -2), (1, 4, -4), (-1, 4, -4).
Equations multiplied by -1 do not have a different solution set; for example,  (-7, -1, 1) has the same solution set as (7, 1, -1).
Equations with GCD(u, v, w) != 1 are excluded, because their solution sets are assigned to equations with lower n. For example, (3, 3, -3) is not included here, because its solution set is already assigned to (1, 1, -1).
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; seq(A365876(n), n = 1 .. 74);

Prog

(Python)
from math import gcd
def A365876(n):
    if n == 1: return 1
    c = 0
    for v in range(1, n+1>>1):
        u = n-(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 04 2023

Crossrefs

Cf. A364384, A364385, A365877 (partial sums), A365892.

Sequence in context: A308308 A024162 A334677 * A179080 A294199 A078658

Adjacent sequences: A365873 A365874 A365875 * A365877 A365878 A365879

Keyword

nonn

Author

Felix Huber, Sep 22 2023

Status

approved