A377607 Positive integers D such that the generalized Pell equation X^2 - D Y^2 = 3 is solvable over the integers.

1, 6, 13, 22, 33, 46, 61, 69, 73, 78, 94, 97, 109, 118, 141, 157, 166, 177, 181, 193, 213, 214, 222, 241, 249, 253, 262, 277, 286, 313, 321, 334, 337, 358, 366, 382, 393, 397, 409, 421, 429, 433, 438, 454, 457, 478, 481, 501, 502, 517, 526, 537, 541, 573, 598, 601, 613, 622, 649, 654, 661

Offset

1,2

Comments

Calculated using Dario Alpern's quadratic Diophantine solver, see link.

Links

Robin Visser, Table of n, a(n) for n = 1..10000

Dario Alpern, Generic two integer variable equation solver.

Eric Weisstein's World of Mathematics, Pell Equation.

Example

The first fundamental solutions [x(n), y(n)] are (the first entry gives D(n)=a(n)):
[1, [2, 1]], [6, [3, 1]], [13, [4, 1]], [22, [5, 1]], [33, [6, 1]], [46, [7, 1]], [61, [8, 1]], [69, [108, 13]], [73, [94, 11]], [78, [9, 1]], [94, [223, 23]], [97, [10, 1]], [109, [9532, 913]], [118, [11, 1]], [141, [12, 1]], [157, [289580, 23111]], [166, [13, 1]], [177, [306, 23]], [181, [148, 11]], [193, [14, 1]], ...

Prog

(Python)
from itertools import count, islice
from sympy.solvers.diophantine.diophantine import diop_DN
def A377607_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda d:len(diop_DN(d, 3)), count(max(startvalue, 1)))
A377607_list = list(islice(A377607_gen(), 61)) # Chai Wah Wu, Nov 03 2024

Crossrefs

Cf. A031396, A243655, A261246, A377600.

Sequence in context: A173358 A101247 A243655 * A072212 A028872 A049718

Adjacent sequences: A377604 A377605 A377606 * A377608 A377609 A377610

Keyword

nonn

Author

Robin Visser, Nov 02 2024

Status

approved