A337779 Odd composite integers m such that U(m)^2 == 1 (mod m) and V(m) == 5 (mod m), where U(m)=A004254(m) and V(m)=A003501(m) are the m-th generalized Lucas and Pell-Lucas numbers of parameters a=5 and b=1, respectively.

527, 551, 1105, 1807, 1919, 2015, 2071, 2915, 3289, 4031, 4033, 4355, 5291, 5777, 5983, 6049, 6061, 6479, 6785, 7645, 8695, 9361, 9889, 11285, 11663, 11951, 12209, 12265, 12545, 13079, 14491, 16211, 17119, 17249, 18299, 18407, 20087, 20099, 20845, 21505, 22499

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=5 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, 10000, 2], CompositeQ[#] && Divisible[2*ChebyshevT[#, 5/2] - 5, #] && Divisible[ChebyshevU[#-1, 5/2]*ChebyshevU[#-1, 5/2] - 1, #] &]

Prog

(Python)
from itertools import islice
from sympy import nextprime
def A337779_gen(): # generator of terms
    def f(n, m): # A004254(n) mod m and A003501(n) mod m
        a, b, c, d, a2, b2, c2, d2 = 5, -1, 1, 0, 1, 0, 0, 1
        for x in bin(n)[2:]:
            e, f = b2*c2%m, k if (k:=a2+d2)<m else k-m
            a2, b2, c2, d2 = (a2*a2+e)%m, b2*f%m, c2*f%m, (d2*d2+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, (5*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==5:
                yield m
        p, q = q+2, nextprime(q)
A337779_list = list(islice(A337779_gen(), 30)) # Chai Wah Wu, Jul 07 2026

Crossrefs

Cf. A337628 (a=5, b=-1), A337778 (a=4, b=1).

Sequence in context: A264804 A093226 A153660 * A261075 A389148 A250754

Adjacent sequences: A337776 A337777 A337778 * A337780 A337781 A337782

Keyword

nonn,changed

Author

Ovidiu Bagdasar, Sep 20 2020

Extensions

More terms from Amiram Eldar, Sep 21 2020

Status

approved