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A339131
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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.
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8
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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
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OFFSET
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1,1
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COMMENTS
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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).
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REFERENCES
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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)
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LINKS
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MATHEMATICA
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Select[Range[3, 10000, 2], CoprimeQ[#, 45] && CompositeQ[#] && Divisible[LucasL[4*(# - JacobiSymbol[#, 45])] - 2, #] &] (* Amiram Eldar, Nov 26 2020 *)
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CROSSREFS
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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).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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