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A335671
Odd composite integers m such that A087130(m) == 5 (mod m).
6
9, 27, 65, 121, 145, 377, 385, 533, 1035, 1189, 1305, 1885, 2233, 2465, 4081, 5089, 5993, 6409, 6721, 7107, 10877, 11281, 11285, 13281, 13369, 13741, 13833, 14705, 15457, 16721, 17545, 18901, 19601, 19951, 20329, 20705, 22881, 24769, 25345, 26599, 26937, 28741, 29161
OFFSET
1,1
COMMENTS
If p is a prime, then A087130(p)==5 (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=5, b=-1, V(n) recovers A087130(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
9 is the first odd composite integer for which A087130(9)=2744420==5 (mod 9).
MAPLE
M:= <<5|1>, <1|0>>:
f:= proc(n) uses LinearAlgebra:-Modular;
local A;
A:= Mod(n, M, integer[8]);
A:= MatrixPower(n, A, n);
2*A[1, 1] - 5*A[1, 2] mod n;
end proc:
select(t -> f(t) = 5 and not isprime(t), [seq(i, i=3..10^5, 2)]); # Robert Israel, Jun 19 2020
MATHEMATICA
Select[Range[3, 30000, 2], CompositeQ[#] && Divisible[LucasL[#, 5] - 5, #] &] (* Amiram Eldar, Jun 18 2020 *)
CROSSREFS
Cf. A006497, A005845 (a=1), A330276 (a=2), A335669 (a=3), A335670 (a=4).
Sequence in context: A256327 A011923 A029875 * A337628 A129957 A373479
KEYWORD
nonn
AUTHOR
Ovidiu Bagdasar, Jun 17 2020
EXTENSIONS
More terms from Jinyuan Wang, Jun 17 2020
STATUS
approved