A339125 Odd composite integers m such that A000032(m-J(m,5)) == 2*J(m,5) (mod m), where J(m,5) is the Jacobi symbol.

9, 49, 121, 169, 289, 361, 529, 841, 961, 1127, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 3751, 4181, 4489, 4901, 4961, 5041, 5329, 5777, 6241, 6721, 6889, 7381, 7921, 9409, 10201, 10609, 10877, 11449, 11881, 12769, 13201, 15251, 16129, 17161, 18081, 18769, 19321

Offset

1,1

Comments

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-J(p,D)) == 2*J(p,D) (mod p) when p is prime, b=-1 and D=a^2+4.

This sequence has the odd composite integers with V(m-J(m,D)) == 2*J(m,D) (mod m).

For a=1 and b=-1, we have D=5 and V(m) recovers A000032(m) (Lucas numbers).

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)

D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted)

Links

Table of n, a(n) for n=1..44.

Dorin Andrica and Ovidiu Bagdasar, On Generalized Lucas Pseudoprimality of Level k, Mathematics (2021) Vol. 9, 838.

Mathematica

Select[Range[3, 20000, 2], CompositeQ[#] && Divisible[LucasL[# - (j = JacobiSymbol[#, 5])] - 2*j, #] &] (* Amiram Eldar, Nov 26 2020 *)

Crossrefs

Cf. A339126 (a=3, b=-1), A339127 (a=5, b=-1), A339128 (a=7, b=-1), A339129 (a=3, b=1), A339130 (a=5, b=1), A339131 (a=7, b=1).

Sequence in context: A369963 A012111 A138998 * A016838 A087691 A014730

Adjacent sequences: A339122 A339123 A339124 * A339126 A339127 A339128

Keyword

nonn

Author

Ovidiu Bagdasar, Nov 24 2020

Status

approved