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A339129
Odd composite integers m such that A005248(m-J(m,5)) == 2 (mod m), where J(m,5) is the Jacobi symbol.
8
9, 49, 63, 121, 169, 289, 323, 361, 377, 441, 529, 841, 961, 1127, 1369, 1681, 1849, 1891, 2209, 2303, 2809, 2961, 3481, 3721, 3751, 3827, 4181, 4489, 4901, 4961, 5041, 5329, 5491, 5777, 6137, 6241, 6601, 6721, 6889, 7381, 7921, 8149, 9409, 10201, 10609, 10877, 10933, 11449, 11663
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 (mod p) when p is prime, b=1 and D=a^2-4.
This sequence contains the odd composite integers with V(m-J(m,D)) == 2 (mod m).
For a=3 and b=1, we have D=5 and V(m) recovers A005248(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, 12000, 2], CompositeQ[#] && Divisible[LucasL[2*(# - JacobiSymbol[#, 5])] - 2, #] &] (* Amiram Eldar, Nov 26 2020 *)
CROSSREFS
Cf. A005248.
Cf. A339125 (a=1, b=-1), A339126 (a=3, b=-1), A339127 (a=5, b=-1), A339128 (a=7, b=-1), A339130 (a=5, b=1), A339131 (a=7, b=1).
Sequence in context: A012260 A262537 A133478 * A340239 A032589 A137175
KEYWORD
nonn
AUTHOR
Ovidiu Bagdasar, Nov 24 2020
STATUS
approved