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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.
8
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
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
KEYWORD
nonn
AUTHOR
Ovidiu Bagdasar, Nov 24 2020
STATUS
approved