A335669 Odd composite integers m such that A006497(m) == 3 (mod m).

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

Amiram Eldar, Table of n, a(n) for n = 1..500

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

Adjacent sequences: A335666 A335667 A335668 * A335670 A335671 A335672

Keyword

nonn

Author

Ovidiu Bagdasar, Jun 17 2020

Extensions

More terms from Jinyuan Wang, Jun 17 2020

Status

approved