login
A335674
Odd composite integers m such that A003501(m) == 5 (mod m).
4
15, 21, 35, 105, 161, 195, 255, 345, 385, 399, 465, 527, 551, 609, 741, 897, 1105, 1295, 1311, 1807, 1919, 2001, 2015, 2071, 2085, 2121, 2415, 2737, 2915, 3289, 3815, 4031, 4033, 4355, 4879, 4991, 5291, 5777, 5983, 6049, 6061, 6083, 6479, 6601, 6785, 7645, 7905, 8695, 8855, 8911, 9361, 9591, 9889
OFFSET
1,1
COMMENTS
If p is a prime, then A003501(p)==5 (mod p).
This sequence contains the odd composite integers for which the congruence holds.
The generalized Pell-Lucas sequences 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=5, b=1, V(n) recovers A003501(n).
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
15 is the first odd composite integer for which the relation A003501(15)=16098445550==5 (mod 15) holds.
MATHEMATICA
Select[Range[3, 5000, 2], CompositeQ[#] && Divisible[2*ChebyshevT[#, 5/2] - 5, #] &] (* Amiram Eldar, Jun 18 2020 *)
CROSSREFS
Cf. A005248, A335669 (a=3,b=-1), A335672 (a=3,b=1), A335673 (a=4,b=1).
Sequence in context: A081977 A180655 A254365 * A349096 A063175 A367103
KEYWORD
nonn
AUTHOR
Ovidiu Bagdasar, Jun 17 2020
STATUS
approved