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A340095
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Odd composite integers m such that A052918(m-J(m,29)) == 0 (mod m) and gcd(m,29)=1, where J(m,29) is the Jacobi symbol.
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6
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9, 15, 27, 45, 91, 121, 135, 143, 1547, 1573, 1935, 2015, 6543, 6721, 8099, 10403, 10877, 10905, 13319, 13741, 13747, 14399, 14705, 16109, 16471, 18901, 19043, 19109, 19601, 19951, 20591, 22753, 24639, 26599, 26937, 27593
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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 the identity
U(p-J(p,D)) == 0 (mod p) when p is prime, b=-1 and D=a^2+4.
This sequence contains the odd composite integers with U(m-J(m,D)) == 0 (mod m).
For a=5 and b=-1, we have D=29 and U(m) recovers A052918(m).
If even numbers greater than 2 that are coprime to 29 are allowed, then 26, 442, 6994, ... would also be terms. - Jianing Song, Jan 09 2021
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REFERENCES
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D. Andrica and O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
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LINKS
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MATHEMATICA
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Select[Range[3, 28000, 2], CoprimeQ[#, 29] && CompositeQ[#] && Divisible[Fibonacci[#-JacobiSymbol[#, 29], 5], #] &]
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CROSSREFS
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Cf. A052918, A071904, A081264 (a=1, b=-1), A327653 (a=3, b=-1), A340096 (a=7, b=-1), A340097 (a=3, b=1), A340098 (a=5, b=1), A340099 (a=7, b=1).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Coprime condition added to definition by Georg Fischer, Jul 20 2022
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STATUS
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approved
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