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A339130
Odd composite integers m such that A003501(m-J(m,21)) == 2 (mod m) and gcd(m,21)=1, where J(m,21) is the Jacobi symbol.
8
25, 121, 169, 275, 289, 361, 527, 529, 551, 575, 841, 961, 1369, 1681, 1807, 1849, 1919, 2209, 2783, 2809, 3025, 3481, 3721, 4033, 4489, 5041, 5329, 5777, 5983, 6049, 6241, 6479, 6575, 6889, 7267, 7645, 7921, 8959, 8993, 9361, 9409, 9775
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=5 and b=1, we have D=21 and V(m) recovers A003501(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], CoprimeQ[#, 21] && CompositeQ[#] && Divisible[2*ChebyshevT[# - JacobiSymbol[#, 21], 5/2] - 2, #] &] (* Amiram Eldar, Nov 26 2020 *)
CROSSREFS
Cf. A003501.
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), A339131 (a=7, b=1).
Sequence in context: A036057 A083509 A343075 * A256519 A298009 A213445
KEYWORD
nonn
AUTHOR
Ovidiu Bagdasar, Nov 24 2020
STATUS
approved