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A172159
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Nonprime Lucas numbers.
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1
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1, 4, 18, 76, 123, 322, 843, 1364, 5778, 15127, 24476, 39603, 64079, 103682, 167761, 271443, 439204, 710647, 1149851, 1860498, 4870847, 7881196, 12752043, 20633239, 33385282, 87403803, 141422324, 228826127, 599074578, 969323029, 1568397607, 2537720636, 4106118243, 10749957122, 17393796001, 28143753123
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OFFSET
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1,2
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COMMENTS
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A Lucas number L(n) has for multiples L(m) those for which m = n mod 2n.
However, there are composite Lucas numbers with prime indices in the sequence of Lucas numbers; these are coprime to smaller Lucas numbers (such as L(23) and L(29), which have for least prime factor 139 and 59, respectively.
L(m)|L(n) if and only if n = (2k - 1)m, with m > 1 and k > 0 (this is Theorem 16.6 in Koshy's book).
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REFERENCES
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Thomas Koshy, Fibonacci and Lucas Numbers with Applications. New York: John Wiley & Sons Inc. (2001) p. 200
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LINKS
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MATHEMATICA
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Select[LucasL[Range[60]], Not[PrimeQ[#]]&]
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PROG
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(Magma) [Lucas(n): n in [1..60] | not IsPrime(Lucas(n)) ]; // G. C. Greubel, Apr 21 2022
(SageMath) [lucas_number2(n, 1, -1) for n in (1..60) if not is_prime(lucas_number2(n, 1, -1))] # G. C. Greubel, Apr 21 2022
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CROSSREFS
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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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