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A079002
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Numbers n such that the Fibonacci residues F(k) mod n form the complete set (0,1,2,....,n-1).
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5
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1, 2, 3, 4, 5, 6, 7, 9, 10, 14, 15, 20, 25, 27, 30, 35, 45, 50, 70, 75, 81, 100, 125, 135, 150, 175, 225, 243, 250, 350, 375, 405, 500, 625, 675, 729, 750, 875, 1125, 1215, 1250, 1750, 1875, 2025, 2187, 2500, 3125, 3375, 3645, 3750, 4375, 5625, 6075, 6250, 6561
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OFFSET
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1,2
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REFERENCES
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B. Avila and Y. Chen, On moduli for which the Lucas numbers contain a complete residue system, Fibonacci Quarterly, 51 (2013), 151-152.
S. A. Burr, On moduli for which the Fibonacci numbers contain a complete system of residues, Fibonacci Quarterly, 9 (1971), 497-504.
R. L. Graham, D. E. Knuth and O. Patashnick, "Concrete Mathematics", second edition, Addison Wesley, 1994, ex. 6.85, p. 318, p. 562.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..1000
Cheng Lien Lang and Mong Lung Lang, Fibonacci system and residue completeness (2013)
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FORMULA
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Consists of the integers of the form: 5^k, 2*5^k, 4*5^k, 3^j*5^k, 6*5^k, 7*5^k and 14*5^k [see Concrete Mathematics]
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EXAMPLE
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Fibonacci numbers (A000045) are 0,1,1,2,3,5,8,.. and mod 5 these are 0,1,1,2,3,0,3,3,4,... i.e. all possible remainders mod 5 occur in the Fib series mod 5, so 5 is in the series. This is not true for n=8 so 8 is not in the series.
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PROG
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(PARI) is(n)=n/=5^valuation(n, 5); n==3^valuation(n, 3) || setsearch([2, 4, 6, 7, 14], n) \\ Charles R Greathouse IV, Apr 23 2013
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CROSSREFS
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Cf. A066853, A001175, A003593, A224482.
Sequence in context: A060527 A152493 A229028 * A119984 A059879 A132430
Adjacent sequences: A078999 A079000 A079001 * A079003 A079004 A079005
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KEYWORD
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nonn
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AUTHOR
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Benoit Cloitre, Feb 01 2003
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EXTENSIONS
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Corrected by Ron Knott, Jan 05 2005
Entry revised by N. J. A. Sloane, Nov 28 2006, following a suggestion from Martin Fuller
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STATUS
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approved
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