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A339126
Odd composite integers m such that A006497(m-J(m,13)) == 2*J(m,13) (mod m), where J(m,13) is the Jacobi symbol.
8
9, 25, 49, 119, 121, 289, 361, 529, 649, 833, 841, 961, 1089, 1189, 1369, 1681, 1849, 1881, 2023, 2209, 2299, 2809, 3025, 3481, 3721, 4187, 4489, 5041, 5329, 6241, 6889, 7139, 7921, 9409, 10201, 10241, 10609, 11449, 11881, 12769, 12871, 13833, 14041, 14161
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=3 and b=-1, we have D=13 and V(m) recovers A006497(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)
D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted)
MATHEMATICA
Select[Range[3, 15000, 2], CompositeQ[#] && Divisible[LucasL[# - (j = JacobiSymbol[#, 13]), 3] - 2*j, #] &] (* Amiram Eldar, Nov 26 2020 *)
CROSSREFS
Cf. A006497.
Cf. A339125 (a=1, 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: A318737 A246331 A141768 * A176970 A110284 A109367
KEYWORD
nonn
AUTHOR
Ovidiu Bagdasar, Nov 24 2020
STATUS
approved