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A340238
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Odd composite integers m such that A054413(3*m-J(m,53)) == 7 (mod m), where J(m,53) is the Jacobi symbol.
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3
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9, 25, 27, 51, 91, 105, 153, 185, 225, 289, 325, 425, 459, 481, 513, 747, 867, 897, 925, 945, 1001, 1189, 1299, 1469, 1633, 1785, 1921, 2241, 2245, 2599, 2601, 2651, 2769, 2907, 3051, 3277, 3825, 3897, 5681, 6225, 6507, 6777, 7225, 7361, 7803, 8023, 8227, 8701, 8721
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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 (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) (mod m) are called generalized Lucas pseudoprimes of level k- and parameter a.
Here b=-1, a=7, D=53 and k=3, while U(m) is A054413(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[#, 53] && CompositeQ[#] && Divisible[Fibonacci[3*#-JacobiSymbol[#, 53], 7] - 7, #] &]
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CROSSREFS
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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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