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A335673
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Composite integers m such that A003500(m) == 4 (mod m).
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4
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10, 209, 230, 231, 399, 430, 455, 530, 901, 903, 923, 989, 1295, 1729, 1855, 2015, 2211, 2345, 2639, 2701, 2795, 2911, 3007, 3201, 3439, 3535, 3801, 4823, 5291, 5719, 6061, 6767, 6989, 7421, 8569, 9503, 9591, 9869, 9890, 10439, 10609, 11041, 11395, 11951, 11991
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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 A003500(p)==4 (mod p).
This sequence contains the 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.
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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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m=10 is the first composite integer for which A003500(m)==4 (mod m).
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MATHEMATICA
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Select[Range[3, 20000], CompositeQ[#] && Divisible[Round@LucasL[2#, Sqrt[2]] - 4, #] &] (* Amiram Eldar, Jun 18 2020 *)
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PROG
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(PARI) my(M=[1, 2; 1, 3]); forcomposite(m=5, 10^5, if(trace(Mod(M, m)^m)==4, print1(m, ", "))); \\ Joerg Arndt, Jun 18 2020
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CROSSREFS
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A330206 is the subsequence of odd terms.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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