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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.

%I A365877 #17 Oct 05 2023 14:21:01
%S A365877 1,1,2,3,5,6,9,11,15,17,23,25,32,35,40,44,53,56,66,71,78,83,95,100,
%T A365877 111,117,127,134,150,154,171,180,191,199,213,220,240,250,263,272,294,
%U A365877 301,324,335,348,360,386,395,419,430,448,461,490,500,522,536,556,571,603
%N 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.
%C A365877 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.
%H A365877 Wikipedia, <a href="https://en.wikipedia.org/wiki/Quadratic_equation">Quadratic equation</a>.
%H A365877 Wikipedia, <a href="https://en.wikipedia.org/wiki/Vieta%27s_formulas">Vieta's formulas</a>.
%F A365877 a(n) = Sum_{k=1..n} A365876(k).
%F A365877 a(n) = A341123(n) for 1 <= n <= 13.
%e A365877 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).
%e A365877 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).
%e A365877 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).
%e A365877 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.
%p A365877 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;
%p A365877 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);
%o A365877 (Python)
%o A365877 from math import gcd
%o A365877 def A365877(n):
%o A365877     if n == 1: return 1
%o A365877     c = 1
%o A365877     for m in range(2,n+1):
%o A365877         for v in range(1,m+1>>1):
%o A365877             u = m-(v<<1)
%o A365877             if gcd(u,v)==1:
%o A365877                 v2, u2 = v*v, v*(u<<2)
%o A365877                 if v2+u2 >= 0:
%o A365877                     c +=1
%o A365877                 if v2-u2 >= 0:
%o A365877                     c +=1
%o A365877     return c # _Chai Wah Wu_, Oct 05 2023
%Y A365877 Cf. A364384, A364385, A341123, A365892.
%Y A365877 Partial sums of A365876.
%K A365877 nonn
%O A365877 1,3
%A A365877 _Felix Huber_, Sep 22 2023