Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #16 Nov 23 2023 11:47:41
%S 15,21,35,105,161,195,255,345,385,399,465,527,551,609,741,897,1105,
%T 1295,1311,1807,1919,2001,2015,2071,2085,2121,2415,2737,2915,3289,
%U 3815,4031,4033,4355,4879,4991,5291,5777,5983,6049,6061,6083,6479,6601,6785,7645,7905,8695,8855,8911,9361,9591,9889
%N Odd composite integers m such that A003501(m) == 5 (mod m).
%C If p is a prime, then A003501(p)==5 (mod p).
%C This sequence contains the odd composite integers for which the congruence holds.
%C 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.
%C For a=5, b=1, V(n) recovers A003501(n).
%D D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer (to appear, 2020).
%H Chai Wah Wu, <a href="/A335674/b335674.txt">Table of n, a(n) for n = 1..1000</a>
%H D. Andrica and O. Bagdasar, <a href="https://repository.derby.ac.uk/item/92yqq/on-some-new-arithmetic-properties-of-the-generalized-lucas-sequences">On some new arithmetic properties of the generalized Lucas sequences</a>, preprint for Mediterr. J. Math. 18, 47 (2021).
%e 15 is the first odd composite integer for which the relation A003501(15)=16098445550==5 (mod 15) holds.
%t Select[Range[3, 5000, 2], CompositeQ[#] && Divisible[2*ChebyshevT[#, 5/2] - 5, #] &] (* _Amiram Eldar_, Jun 18 2020 *)
%Y Cf. A005248, A335669 (a=3,b=-1), A335672 (a=3,b=1), A335673 (a=4,b=1).
%K nonn
%O 1,1
%A _Ovidiu Bagdasar_, Jun 17 2020