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A340239
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Odd composite integers m such that A001906(3*m-J(m,5)) == 3*J(m,5) (mod m), where J(m,5) is the Jacobi symbol.
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2
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9, 49, 63, 141, 161, 207, 323, 341, 377, 441, 671, 901, 1007, 1127, 1281, 1449, 1853, 1891, 2071, 2303, 2407, 2501, 2743, 2961, 3827, 4181, 4623, 5473, 5611, 5777, 6119, 6593, 6601, 6721, 7161, 7567, 8149, 8473, 8961, 9729, 9881
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OFFSET
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1,1
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COMMENTS
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The generalized Lucas sequences of integer parameters (a,b) defined by U(m+2)=a*U(m+1)-b*U(m) and U(0)=0, U(1)=1, satisfy U(3*p-J(p,D)) == a*J(p,D) (mod p) whenever p is prime, k is a positive integer, b=1 and D=a^2-4.
The composite integers m with the property U(k*m-J(m,D)) == U(k-1)*J(m,D) (mod m) are called generalized Lucas pseudoprimes of level k+ and parameter a.
Here b=1, a=3, D=5 and k=3, while U(m) is A001906(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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Table of n, a(n) for n=1..41.
Dorin Andrica, Vlad Crişan, and Fawzi Al-Thukair, On Fibonacci and Lucas sequences modulo a prime and primality testing, Arab Journal of Mathematical Sciences, 2018, 24(1), 9--15.
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MATHEMATICA
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Select[Range[3, 10000, 2], CoprimeQ[#, 5] && CompositeQ[#] && Divisible[ ChebyshevU[3*# - JacobiSymbol[#, 5] - 1, 3/2] - 3*JacobiSymbol[#, 5], #] &]
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CROSSREFS
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Cf. A001906, A071904, A340097 (a=3, b=1, k=1), A340122 (a=3, b=1, k=2).
Cf. A340240 (a=5, b=1, k=3), A340241 (a=7, b=1, k=3).
Sequence in context: A262537 A133478 A339129 * A032589 A137175 A028375
Adjacent sequences: A340236 A340237 A340238 * A340240 A340241 A340242
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KEYWORD
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nonn
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AUTHOR
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Ovidiu Bagdasar, Jan 01 2021
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STATUS
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approved
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