A079002 Numbers n such that the Fibonacci residues F(k) mod n form the complete set (0,1,2,...,n-1).

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

Offset

1,2

References

R. L. Graham, D. E. Knuth and O. Patashnick, "Concrete Mathematics", second edition, Addison Wesley, 1994, ex. 6.85, p. 318, p. 562.

Links

T. D. Noe, Table of n, a(n) for n = 1..1000

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.

Cheng Lien Lang and Mong Lung Lang, Fibonacci system and residue completeness, arXiv:1304.2892 [math.NT], 2013.

Formula

Consists of the integers of the forms 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].

Example

Fibonacci numbers (A000045) are 0,1,1,2,3,5,8,... and their residues mod 5 are 0,1,1,2,3,0,3,3,4,...; i.e., all possible remainders mod 5 occur in the Fibonacci sequence mod 5, so 5 is in the sequence. This is not true for n=8, so 8 is not in the sequence.

Mathematica

Select[Range[10^4], MatchQ[FactorInteger[#], {{1, 1}}|{{2, 1}}|{{2, 2}}| {{3, _}}|{{2, 1}, {3, 1}}|{{7, 1}}|{{2, 1}, {7, 1}}|{{5, _}}|{{2, 1}, {5, _}}|{{2, 2}, {5, _}}|{{3, _}, {5, _}}|{{2, 1}, {3, 1}, {5, _}}|{{5, _}, {7, 1}}|{{2, 1}, {5, _}, {7, 1}}]&] (* Jean-François Alcover, Sep 01 2018 *)

Prog

(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

Crossrefs

Cf. A066853, A001175, A003593, A224482, A249104.

Sequence in context: A060527 A152493 A229028 * A119984 A059879 A132430

Adjacent sequences: A078999 A079000 A079001 * A079003 A079004 A079005

Keyword

nonn

Author

Benoit Cloitre, Feb 01 2003

Extensions

Corrected by Ron Knott, Jan 05 2005

Entry revised by N. J. A. Sloane, Nov 28 2006, following a suggestion from Martin Fuller

Status

approved