A335673 Composite integers m such that A003500(m) == 4 (mod m).

10, 209, 230, 231, 399, 430, 455, 530, 901, 903, 923, 989, 1295, 1729, 1855, 2015, 2211, 2345, 2639, 2701, 2795, 2911, 3007, 3201, 3439, 3535, 3801, 4823, 5291, 5719, 6061, 6767, 6989, 7421, 8569, 9503, 9591, 9869, 9890, 10439, 10609, 11041, 11395, 11951, 11991

Offset

1,1

Comments

If p is a prime, then A003500(p)==4 (mod p).

This sequence contains the 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=4, b=1, V(n)=A003500(n).

References

D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer (to appear, 2020).

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000 (first 1000 odd terms from Chai Wah Wu)

D. Andrica and O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, preprint for Mediterr. J. Math. 18, 47 (2021).

Example

m=10 is the first composite integer for which A003500(m)==4 (mod m).

Mathematica

Select[Range[3, 20000], CompositeQ[#] && Divisible[Round@LucasL[2#, Sqrt[2]] - 4, #] &] (* Amiram Eldar, Jun 18 2020 *)

Prog

(PARI) my(M=[1, 2; 1, 3]); forcomposite(m=5, 10^5, if(trace(Mod(M, m)^m)==4, print1(m, ", "))); \\ Joerg Arndt, Jun 18 2020

Crossrefs

Cf. A005248, A335669 (a=3,b=-1), A335672 (a=3,b=1), A335674 (a=5,b=1).

A330206 is the subsequence of odd terms.

Sequence in context: A245918 A368441 A211107 * A215555 A069863 A160476

Adjacent sequences: A335670 A335671 A335672 * A335674 A335675 A335676

Keyword

nonn

Author

Ovidiu Bagdasar, Jun 17 2020

Extensions

More terms from Joerg Arndt, Jun 18 2020

Status

approved