A338082 Odd composite integers m such that A056854(m) == 7 (mod m).

9, 15, 21, 35, 45, 63, 99, 105, 195, 231, 315, 323, 329, 369, 377, 423, 435, 451, 595, 665, 705, 805, 861, 903, 1081, 1189, 1443, 1551, 1819, 1833, 1869, 1891, 1935, 2033, 2211, 2345, 2465, 2737, 2849, 2871, 2961, 3059, 3289, 3653, 3689, 3745, 3827, 4005, 4059

Offset: 1

Author

Ovidiu Bagdasar, Oct 08 2020

Keywords

• nonn

Comments

If p is a prime, then A056854(p)==7 (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(m+2)=a*V(m+1)-b*V(m) and V(0)=2, V(1)=a, satisfy the identity V(p)==a (mod p) when p is prime and b=-1,1.

For a=7 and b=1, V(m) recovers A056854(m).

References

• D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
• D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021)

Crossrefs

Cf. A005248, A335669 (a=3,b=-1), A335672 (a=3,b=1), A335673 (a=4,b=1), A335674 (a=5,b=1), A337233 (a=6,b=1).

Programs

Mathematica

Select[Range[3, 20000, 2], CompositeQ[#] && Divisible[2*ChebyshevT[#, 7/2] - 7, #] &]
Select[Range[9,5001,2],CompositeQ[#]&&Mod[LucasL[4#],#]==7&] (* Harvey P. Dale, Apr 28 2022 *)