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A339131 Odd composite integers m such that A056854(m-J(m,45)) == 2 (mod m) and gcd(m,45)=1, where J(m,45) is the Jacobi symbol. 8
49, 121, 169, 289, 323, 329, 361, 377, 451, 529, 841, 961, 1081, 1127, 1369, 1681, 1819, 1849, 1891, 2033, 2209, 2303, 2809, 3481, 3653, 3721, 3751, 3827, 4181, 4489, 4901, 4961, 5041, 5329, 5491, 5671, 5777, 6137, 6241, 6601, 6721, 6889, 7381, 7921, 8149, 8557, 9409 (list; graph; refs; listen; history; text; internal format)
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=7 and b=1, we have D=45 and 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)
D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted)
LINKS
MATHEMATICA
Select[Range[3, 10000, 2], CoprimeQ[#, 45] && CompositeQ[#] && Divisible[LucasL[4*(# - JacobiSymbol[#, 45])] - 2, #] &] (* Amiram Eldar, Nov 26 2020 *)
CROSSREFS
Cf. A056854.
Cf. A339125 (a=1, b=-1), 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).
Sequence in context: A084733 A338499 A227863 * A374290 A115557 A167718
KEYWORD
nonn
AUTHOR
Ovidiu Bagdasar, Nov 24 2020
STATUS
approved

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Last modified July 19 10:32 EDT 2024. Contains 374392 sequences. (Running on oeis4.)