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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.
5
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.
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
KEYWORD
nonn
AUTHOR
Felix Huber, Sep 22 2023
STATUS
approved