A337782 Even composite integers m such that U(m)^2 == 1 (mod m) and V(m) == 7 (mod m), where U(m)=A004187(m) and V(m)=A056854(m) are the m-th generalized Lucas and Pell-Lucas numbers of parameters a=7 and b=1, respectively.

4, 8, 44, 104, 136, 152, 232, 286, 442, 836, 1364, 1378, 2204, 2584, 2626, 2684, 2834, 3016, 3926, 4636, 5662, 7208, 7384, 7676, 7964, 8294, 9164, 9316, 11476, 12524, 14824, 15224, 17324, 20026, 20474, 21736, 21944, 22814, 23804, 24616, 26596, 27028, 27404, 31124

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..8220

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[2, 20000, 2], CompositeQ[#] && Divisible[2*ChebyshevT[#, 7/2] - 7, #] && Divisible[ChebyshevU[#-1, 7/2]*ChebyshevU[#-1, 7/2] - 1, #] &]

Prog

(Python)
from itertools import islice
from sympy import nextprime
def A337782_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 = 4, 5
    while True:
        for m in range(p, q, 2):
            a, b = f(m, m)
            if pow(a, 2, m)==1 and b==7%m:
                yield m
        p, q = q+1, nextprime(q)
A337782_list = list(islice(A337782_gen(), 40)) # Chai Wah Wu, Jul 01 2026

Crossrefs

Cf. A337630 (a=7, b=-1), A337777 (a=3, b=1), A337781 (a=7, b=1).

Sequence in context: A163343 A284972 A045639 * A285751 A189538 A351071

Adjacent sequences: A337779 A337780 A337781 * A337783 A337784 A337785

Keyword

nonn,changed

Author

Ovidiu Bagdasar, Sep 20 2020

Extensions

More terms from Amiram Eldar, Sep 21 2020

Status

approved