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

Author

Ovidiu Bagdasar, Jun 17 2020

Keywords

• nonn

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).

Examples

			33 is the first odd composite integer for which we have A006497(33) = 132742316047301964 == 3 (mod 33).
		

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).

Crossrefs

Cf. A006497, A005845 (a=1), A330276 (a=2), A335670 (a=4), A335671 (a=5).

Programs

Mathematica

Select[Range[3, 50000, 2], CompositeQ[#] && Divisible[LucasL[#, 3] - 3, #] &] (* Amiram Eldar, Jun 18 2020 *)

PARI

is(m) = m%2 && !isprime(m) && [2, 3]*([0, 1; 1, 3]^m)[, 1]%m==3; \\ Jinyuan Wang, Jun 17 2020

Extensions

More terms from Jinyuan Wang, Jun 17 2020