A335672 Odd composite integers m such that A005248(m) == 3 (mod m).

15, 105, 195, 231, 323, 377, 435, 665, 705, 1443, 1551, 1891, 2465, 2737, 2849, 3289, 3689, 3745, 3827, 4181, 4465, 4879, 5655, 5777, 6479, 6601, 6721, 7055, 7743, 8149, 9879, 10815, 10877, 11305, 11395, 11663, 12935, 13201, 13981, 15251, 15301, 17119, 17261, 17711, 18407, 18915, 19043, 20999

Offset

1,1

Comments

If p is a prime, then A005248(p)==3 (mod p).

This sequence contains the odd 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=3, b=1, V(n) recovers A005248(n) (bisection of Fibonacci numbers).

References

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

Links

Chai Wah Wu, 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

15 is the first odd composite integer for which A005248(15)=18604984==3 (mod 15).

Mathematica

Select[Range[3, 10000, 2], CompositeQ[#] && Divisible[LucasL[2#] - 3, #] &] (* Amiram Eldar, Jun 18 2020 *)

Crossrefs

Cf. A005248, A335669 (a=3,b=-1), A335673 (a=4,b=1), A335674 (a=5,b=1).

Sequence in context: A223441 A300295 A102791 * A160892 A061550 A174385

Adjacent sequences: A335669 A335670 A335671 * A335673 A335674 A335675

Keyword

nonn

Author

Ovidiu Bagdasar, Jun 17 2020

Status

approved