A337625 Odd composite integers m such that F(m)^2 == 1 (mod m) and L(m) == 1 (mod m), where F(m) and L(m) are the m-th Fibonacci and Lucas numbers, respectively.

2737, 4181, 5777, 6721, 10877, 13201, 15251, 29281, 34561, 51841, 64079, 64681, 67861, 68251, 75077, 80189, 90061, 96049, 97921, 100127, 105281, 113573, 118441, 146611, 161027, 162133, 163081, 179697, 186961, 194833, 197209, 219781, 228241, 231703, 252601, 254321

Offset

1,1

Comments

Intersection of A005845 and A337231.

These numbers may be called weak generalized Fibonacci-Lucas-Bruckner pseudoprimes.

If p is a prime, then F(p)^2 == 1 (mod p) and L(p) == 1 (mod p).

This sequence contains the odd composite integers for which these congruences hold.

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=1 and b=-1.

Links

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

Dorin Andrica and Ovidiu Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math., 18 (2021), 47.

Mathematica

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

Prog

(Python)
from itertools import islice
from sympy import nextprime
def A337625_gen(): # generator of terms
    def f(n): # fibonacci(n) and lucas(n) mod n
        a, b, c, d, a2, b2, c2, d2 = 1, 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 b2, (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==1:
                yield m
        p, q = q+2, nextprime(q)
A337625_list = list(islice(A337625_gen(), 40)) # Chai Wah Wu, Jul 01 2026

Crossrefs

Cf. A005845 and A337231.

Sequence in context: A253987 A258904 A221976 * A094401 A035774 A107570

Adjacent sequences: A337622 A337623 A337624 * A337626 A337627 A337628

Keyword

nonn

Author

Ovidiu Bagdasar, Sep 19 2020

Extensions

More terms from Amiram Eldar, Sep 19 2020

Status

approved