A335671 Odd composite integers m such that A087130(m) == 5 (mod m).

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

Robert Israel, Table of n, a(n) for n = 1..1000

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 A328408

Adjacent sequences: A335668 A335669 A335670 * A335672 A335673 A335674

Keyword

nonn

Author

Ovidiu Bagdasar, Jun 17 2020

Extensions

More terms from Jinyuan Wang, Jun 17 2020

Status

approved