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A335674
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Odd composite integers m such that A003501(m) == 5 (mod m).
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4
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15, 21, 35, 105, 161, 195, 255, 345, 385, 399, 465, 527, 551, 609, 741, 897, 1105, 1295, 1311, 1807, 1919, 2001, 2015, 2071, 2085, 2121, 2415, 2737, 2915, 3289, 3815, 4031, 4033, 4355, 4879, 4991, 5291, 5777, 5983, 6049, 6061, 6083, 6479, 6601, 6785, 7645, 7905, 8695, 8855, 8911, 9361, 9591, 9889
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OFFSET
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1,1
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COMMENTS
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If p is a prime, then A003501(p)==5 (mod p).
This sequence contains the odd composite integers for which the congruence holds.
The generalized Pell-Lucas sequences of integer parameters (a,b) defined by V(n+2)=a*V(n+1)-b*V(n) and V(0)=2, V(1)=a, satisfy the identity V(p)==a (mod p) whenever p is prime and b=-1,1.
For a=5, b=1, V(n) recovers A003501(n).
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REFERENCES
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D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer (to appear, 2020).
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LINKS
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EXAMPLE
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15 is the first odd composite integer for which the relation A003501(15)=16098445550==5 (mod 15) holds.
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MATHEMATICA
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Select[Range[3, 5000, 2], CompositeQ[#] && Divisible[2*ChebyshevT[#, 5/2] - 5, #] &] (* Amiram Eldar, Jun 18 2020 *)
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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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