login
A335669
Odd composite integers m such that A006497(m) == 3 (mod m).
10
33, 65, 119, 273, 377, 385, 533, 561, 649, 1105, 1189, 1441, 2065, 2289, 2465, 2849, 4187, 4641, 6545, 6721, 11921, 12871, 13281, 14041, 15457, 16109, 18241, 19201, 22049, 23479, 24769, 25345, 28421, 30745, 31631, 34997, 38121, 38503, 41441, 45961, 46761, 48577
OFFSET
1,1
COMMENTS
If p is a prime, then A006497(p) == 3 (mod p).
This sequence contains the odd composite integers for which the congruence holds.
The generalized Pell-Lucas sequence of integer parameters (a,b) defined by V(n+2)=a*V(n+1)-b*V(n) and V(0)=2, V(1)=a, satisfy the identity V(p)==a (mod p) whenever p is prime and b=-1,1.
For a=3, b=-1, V(n) recovers the sequence A006497(n) (bronze Fibonacci numbers).
REFERENCES
D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer (to appear, 2020).
LINKS
D. Andrica and O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, preprint for Mediterr. J. Math. 18, 47 (2021).
EXAMPLE
33 is the first odd composite integer for which we have A006497(33) = 132742316047301964 == 3 (mod 33).
MATHEMATICA
Select[Range[3, 50000, 2], CompositeQ[#] && Divisible[LucasL[#, 3] - 3, #] &] (* Amiram Eldar, Jun 18 2020 *)
PROG
(PARI) is(m) = m%2 && !isprime(m) && [2, 3]*([0, 1; 1, 3]^m)[, 1]%m==3; \\ Jinyuan Wang, Jun 17 2020
CROSSREFS
Cf. A006497, A005845 (a=1), A330276 (a=2), A335670 (a=4), A335671 (a=5).
Sequence in context: A061560 A118618 A163411 * A339908 A071595 A225707
KEYWORD
nonn
AUTHOR
Ovidiu Bagdasar, Jun 17 2020
EXTENSIONS
More terms from Jinyuan Wang, Jun 17 2020
STATUS
approved