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

Author

Ovidiu Bagdasar, Jun 17 2020

Keywords

• nonn

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).

Examples

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

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).

Crossrefs

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

Programs

Mathematica

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