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A339128
Odd composite integers m such that A086902(m-J(m,53)) == 2*J(m,53) (mod m), where J(m,53) is the Jacobi symbol.
8
9, 25, 49, 51, 91, 121, 125, 153, 169, 289, 325, 361, 441, 529, 625, 637, 833, 841, 867, 961, 1183, 1225, 1369, 1633, 1681, 1849, 1921, 2209, 2599, 2601, 2651, 3481, 3721, 4225, 4489, 4625, 5041, 5125, 5329, 5537, 6241, 6889, 7225, 7267, 7497, 7921, 8125, 8281
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=7 and b=-1, we have D=53 and V(m) recovers A086902(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, 10000, 2], CompositeQ[#] && Divisible[LucasL[# - (j = JacobiSymbol[#, 53]), 7] - 2*j, #] &] (* Amiram Eldar, Nov 26 2020 *)
CROSSREFS
Cf. A086902.
Cf. A339125 (a=1, b=-1), A339126 (a=3, b=-1), A339127 (a=5, b=-1), A339129 (a=3, b=1), A339130 (a=5, b=1), A339131 (a=7, b=1).
Sequence in context: A247687 A075026 A339727 * A113659 A325701 A113745
KEYWORD
nonn
AUTHOR
Ovidiu Bagdasar, Nov 24 2020
EXTENSIONS
More terms from Amiram Eldar, Nov 26 2020
STATUS
approved