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Odd composite integers m such that A003501(m) == 5 (mod m).
4

%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