A337626 Odd composite integers m such that U(m)^2 == 1 (mod m) and V(m) == 3 (mod m), where U(m) and V(m) are the m-th generalized Lucas and Pell-Lucas numbers of parameters a=3 and b=-1, respectively.

33, 119, 385, 561, 649, 1189, 1441, 2065, 2289, 2465, 2849, 4187, 6545, 12871, 13281, 14041, 16109, 18241, 22049, 23479, 24769, 25345, 28421, 31631, 34997, 38121, 38503, 41441, 45961, 48577, 50545, 53585, 56279, 58081, 59081, 61447, 63393, 66385, 75077, 91187

Offset

1,1

Comments

Intersection of A335669 and A337234.

For a,b integers, the following sequences are defined:

* generalized Lucas sequences by U(n+2)=a*U(n+1)-b*U(n) and U(0)=0, U(1)=1,

* generalized Pell-Lucas sequences by V(n+2)=a*V(n+1)-b*V(n) and V(0)=2, V(1)=a.

These satisfy the identities U(p)^2 == 1 and V(p)==a (mod p) for p prime and b=1,-1.

These numbers may be called weak generalized Lucas-Bruckner pseudoprimes of parameters a and b.The current sequence is defined for a=3 and b=-1.

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000

D. Andrica and O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, preprint for Mediterr. J. Math. 18, 47 (2021).

Mathematica

Select[Range[3, 20000, 2], CompositeQ[#] && Divisible[Fibonacci[#, 3]*Fibonacci[#, 3] - 1, #] && Divisible[LucasL[#, 3] - 3, #] &]

Prog

(Python)
from itertools import islice
from sympy import nextprime
def A337626_gen(): # generator of terms
    def f(n):
        a, b, c, d, a2, b2, c2, d2 = 3, 1, 1, 0, 1, 0, 0, 1
        for x in bin(n)[2:]:
            e, f = b2*c2%n, a2+d2
            a2, b2, c2, d2 = (pow(a2, 2, n)+e)%n, b2*f%n, c2*f%n, (pow(d2, 2, n)+e)%n
            if x=='1':
                a2, b2, c2, d2 = (a2*a+b2*c)%n, (a2*b+b2*d)%n, (c2*a+d2*c)%n, (c2*b+d2*d)%n
        return c2, (3*c2+(d2<<1))%n
    p, q = 9, 11
    while True:
        for m in range(p, q, 2):
            a, b = f(m)
            if pow(a, 2, m)==1 and b==3:
                yield m
        p, q = q+2, nextprime(q)
A337626_list = list(islice(A337626_gen(), 40)) # Chai Wah Wu, Jul 01 2026

Crossrefs

Cf. A335669, A337234.

Sequence in context: A044665 A140161 A177211 * A397165 A301633 A039440

Adjacent sequences: A337623 A337624 A337625 * A337627 A337628 A337629

Keyword

nonn

Author

Ovidiu Bagdasar, Sep 19 2020

Extensions

More terms from Amiram Eldar, Sep 19 2020

Status

approved