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%I #41 Sep 30 2019 03:14:10
%S 1,2,3,4,5,6,7,9,10,14,15,20,25,27,30,35,45,50,70,75,81,100,125,135,
%T 150,175,225,243,250,350,375,405,500,625,675,729,750,875,1125,1215,
%U 1250,1750,1875,2025,2187,2500,3125,3375,3645,3750,4375,5625,6075,6250,6561
%N Numbers n such that the Fibonacci residues F(k) mod n form the complete set (0,1,2,...,n-1).
%D R. L. Graham, D. E. Knuth and O. Patashnick, "Concrete Mathematics", second edition, Addison Wesley, 1994, ex. 6.85, p. 318, p. 562.
%H T. D. Noe, <a href="/A079002/b079002.txt">Table of n, a(n) for n = 1..1000</a>
%H B. Avila and Y. Chen, <a href="https://www.fq.math.ca/Papers1/51-2/AvilaChen.pdf">On moduli for which the Lucas numbers contain a complete residue system</a>, Fibonacci Quarterly, 51 (2013), 151-152.
%H S. A. Burr, <a href="http://www.fq.math.ca/Scanned/9-5/burr.pdf">On moduli for which the Fibonacci numbers contain a complete system of residues</a>, Fibonacci Quarterly, 9 (1971), 497-504.
%H Cheng Lien Lang and Mong Lung Lang, <a href="http://arxiv.org/abs/1304.2892">Fibonacci system and residue completeness</a>, arXiv:1304.2892 [math.NT], 2013.
%F 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].
%e 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.
%t 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 *)
%o (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
%Y Cf. A066853, A001175, A003593, A224482, A249104.
%K nonn
%O 1,2
%A _Benoit Cloitre_, Feb 01 2003
%E Corrected by _Ron Knott_, Jan 05 2005
%E Entry revised by _N. J. A. Sloane_, Nov 28 2006, following a suggestion from _Martin Fuller_
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