A365892 a(n) is the index of the n-th term of A365876 that includes at least one equation with at least one integer solution.

1, 4, 9, 11, 20, 22, 35, 37, 54, 56, 77, 79, 104, 106, 135, 137, 170, 172, 209, 211, 252, 254, 299, 301, 350, 352, 405, 407, 464, 466, 527, 529, 594, 596, 665, 667, 740, 742, 819, 821, 902, 904, 989, 991, 1080, 1082, 1175, 1177, 1274, 1276, 1377, 1379, 1484, 1486

Offset: 1

Author

Felix Huber, Sep 22 2023

Keywords

• nonn

Comments

Observation (checked up to a(52)): a(n) = A266257(n) for n >= 2.

Conjectures in formula section hold for 2<=n<=300. - Chai Wah Wu, Oct 05 2023

Examples

			a(1) = 1 since the equation x^2 = 0 belonging to A365876(1) has the integer solution 0. 1 is the 1st term that includes at least one equation with at least one integer solution.
a(2) = 4 since the equation 2*x^2 + x - 1 = 0 belonging to A365876(4) has the integer solution -1. 4 is the 2nd term that includes at least one equation with at least one integer solution.
a(3) = 9 since the equation -x^2 + 4*x - 1 = 0 belonging to A365876(9) has the integer solution 2. 9 is the 3rd term that includes at least one equation with at least one integer solution.
a(4) = 11 since the equation 3*x^2 + 4*x - 4 = 0 belonging to A365876(11) has the integer solution -2. 11 is the 4th term that includes at least one equation with at least one integer solution.
		

Crossrefs

Cf. A364384, A364385, A365876, A365877.

Programs

Maple

A365892 := proc(n_A365876) local u, v, a, min, x_1, x_2; u := n_A365876; v := 0; a := false; min := true; while min = true do if u <> 0 and gcd(u, v) = 1 then x_1 := 1/2*(-v + sqrt(v^2 + 4*v*u))/u; x_2 := 1/2*(-v - sqrt(v^2 + 4*v*u))/u; if x_1 = floor(x_1) or x_2 = floor(x_2) then a := true; end if; end if; u := u - 2; v := 1/2*n_A365876 - 1/2*abs(u); if u < -1/9*n_A365876 then min := false; end if; end do; if a = true then return n_A365876; end if; end proc; seq(A365892(n_A365876), n_A365876 = 1 .. 1486);

Python

from math import gcd
from itertools import count, islice
from sympy import integer_nthroot
def A365892_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        if n == 1:
            yield 1
        else:
            for v in range(1,n+1>>1):
                u = n-(v<<1)
                if gcd(u,v)==1:
                    v2, u2, a = v*v, v*(u<<2), u<<1
                    if v2+u2 >= 0:
                        d,r = integer_nthroot(v2+u2,2)
                        if r and not ((d+v)%a and (d-v)%a):
                            yield n
                            break
                    if v2-u2 >= 0:
                        d,r = integer_nthroot(v2-u2,2)
                        if r and not ((d+v)%a and (d-v)%a):
                            yield n
                            break
A365892_list = list(islice(A365892_gen(),20)) # Chai Wah Wu, Oct 04 2023

Formula

Conjectures (see also A266257): (Start)

a(1) = 1, a(n) = ((n + 1)^2 - (-1)^n*(n - 1))/2 for n >= 2.

a(1) = 1, a(2) = 4, a(3) = 9, a(4) = 11, a(5) = 20, a(6) = 22, a(n) = a(n - 1) + 2*a(n - 2) - 2*a(n - 3) - a(n - 4) + a(n - 5) for n >= 7.

G.f.: (1 + x + 3*x^3 - x^4)/((1 - x)^3*(1 + x)^2). (End)