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

1,1

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)

Links

Table of n, a(n) for n=1..49.

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 *)

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).

Sequence in context: A020192 A241809 A063174 * A336115 A072569 A072572

Adjacent sequences: A338079 A338080 A338081 * A338083 A338084 A338085

Keyword

nonn

Author

Ovidiu Bagdasar, Oct 08 2020

Status

approved