A337630 Odd composite integers m such that U(m)^2 == 1 (mod m) and V(m) == 7 (mod m), where U(m) and V(m) are the m-th generalized Lucas and Pell-Lucas numbers of parameters a=7 and b=-1, respectively.

25, 51, 91, 161, 325, 425, 561, 791, 1105, 1633, 1921, 2001, 2465, 2599, 2651, 2737, 7345, 8449, 9361, 10325, 10465, 10825, 11285, 12025, 12291, 13021, 15457, 17111, 18193, 18881, 19307, 20705, 20833, 21931, 24081, 24661, 31521, 32305, 37925, 38801, 39059, 40641

Offset

1,1

Comments

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=7 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, 15000, 2], CompositeQ[#] && Divisible[Fibonacci[#, 7]*Fibonacci[#, 7] - 1, #] && Divisible[LucasL[#, 7] - 7, #] &]

Prog

(Python)
from itertools import islice
from sympy import nextprime
def A337630_gen(): # generator of terms
    def f(n, m): # A004187(n) mod m and A056854(n) mod m
        a, b, c, d, a2, b2, c2, d2 = 7, 1, 1, 0, 1, 0, 0, 1
        for x in bin(n)[2:]:
            e, f = b2*c2%m, a2+d2
            a2, b2, c2, d2 = (pow(a2, 2, m)+e)%m, b2*f%m, c2*f%m, (pow(d2, 2, m)+e)%m
            if x=='1':
                a2, b2, c2, d2 = (a2*a+b2*c)%m, (a2*b+b2*d)%m, (c2*a+d2*c)%m, (c2*b+d2*d)%m
        return c2, (7*c2+(d2<<1))%m
    p, q = 9, 11
    while True:
        for m in range(p, q, 2):
            a, b = f(m, m)
            if pow(a, 2, m)==1 and b==7:
                yield m
        p, q = q+2, nextprime(q)
A337630_list = list(islice(A337630_gen(), 42)) # Chai Wah Wu, Jul 01 2026

Crossrefs

Cf. A337625 (a=1), A337626 (a=3), A337627 (a=4), A337628 (a=5), A337629 (a=6).

Sequence in context: A042232 A273868 A338079 * A270693 A042240 A042242

Adjacent sequences: A337627 A337628 A337629 * A337631 A337632 A337633

Keyword

nonn,changed

Author

Ovidiu Bagdasar, Sep 19 2020

Extensions

More terms from Amiram Eldar, Sep 19 2020

Status

approved