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Search: id:A081264
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| A081264 |
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Odd Fibonacci pseudoprimes: odd numbers n such that either (1) n divides Fibonacci(n-1) if n mod 5 = 1 or -1 or (2) n divides Fibonacci(n+1) if n mod 5 = 2 or -2. |
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+0
4
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| 323, 377, 1891, 3827, 4181, 5777, 6601, 6721, 8149, 10877, 11663, 13201, 13981, 15251, 17119, 17711, 18407, 19043, 23407, 25877, 27323, 30889, 34561, 34943, 35207, 39203, 40501, 50183, 51841, 51983, 52701, 53663, 60377, 64079, 64681
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Lehmer shows that there are an infinite number of Fibonacci pseudoprimes (FPPs). In particular, the number Fibonacci(2p) is an FPP for all primes p > 5. Anderson lists over 5000 FPPs. The sequences A069106 and A069107 give n such that n divides Fibonacci(n-1) and n divides Fibonacci(n+1), respectively. See A141137 for even FPPs.
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REFERENCES
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R. Crandall and C. Pomerance, Primes Numbers: A Computational Perspective, Springer, 2002, p. 131.
E. Lehmer, On the infinitude of Fibonacci pseudoprimes, Fibonacci Quarterly, 2, 1964, pp. 229-230.
P. Ribenboim, The New Book of Prime Number Records, Springer, 1995, p. 127.
A. Witno, Theory of Numbers, BookSurge, North Charleston, SC; see p. 83.
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LINKS
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P. G. Anderson, Table of n, a(n) for n = 1..5861 [taken from link below]
P. G. Anderson, Fibonacci pseudoprimes under 2,217,967,487 and their factors
Eric Weisstein's World of Mathematics, Fibonacci Pseudoprime
Wikipedia, Fibonacci pseudoprime
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MATHEMATICA
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lst={}; f0=0; f1=1; Do[f2=f1+f0; If[n>1&&!PrimeQ[n], If[MemberQ[{1, 4}, Mod[n, 5]], If[Mod[f0, n]==0, AppendTo[lst, n]]]; If[MemberQ[{2, 3}, Mod[n, 5]], If[Mod[f2, n]==0, AppendTo[lst, n]]]]; f0=f1; f1=f2, {n, 100000}]; lst
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CROSSREFS
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Cf. A069106, A069107.
Sequence in context: A033524 A082947 A082948 this_sequence A069107 A094412 A065822
Adjacent sequences: A081261 A081262 A081263 this_sequence A081265 A081266 A081267
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KEYWORD
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nice,nonn
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AUTHOR
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T. D. Noe (noe(AT)sspectra.com), Mar 15 2003, Jun 09 2008
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