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A189761 Numbers n for which the set of residues {Fibonacci(k) mod n, k=0,1,2,....} is minimal.

%I A189761 #16 Dec 11 2019 09:49:46
%S A189761 1,2,3,4,5,8,11,21,29,55,76,144,199,377,521,987,1364,2584,3571,6765,
%T A189761 9349,17711,24476,46368,64079,121393,167761,317811,439204,832040,
%U A189761 1149851,2178309,3010349,5702887,7881196,14930352,20633239,39088169,54018521,102334155
%N A189761 Numbers n for which the set of residues {Fibonacci(k) mod n, k=0,1,2,....} is minimal.
%C A189761 Sequence A066853 gives the number of possible residues of the Fibonacci numbers mod n. For the n in this sequence, it appears that A066853(n) < A066853(m) for all m > n. For these n, the set of residues consists of Fibonacci numbers < n and some of their negatives (see example).
%C A189761 Interestingly, for n > 5, this sequence alternates the even-index Fibonacci and odd-indexed Lucas numbers, A001906 and A002878. See A109794 for the sequence without 2 and 5.
%C A189761 The number of residues is 1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 13, 15, 16, ..., which is A032766 with 2 and 5 included.
%H A189761 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-1).
%F A189761 From _Colin Barker_, Oct 29 2013: (Start)
%F A189761 a(n) = 3*a(n-2) - a(n-4) for n > 9.
%F A189761 G.f.: x*(x^8 + x^7 - x^6 - 2*x^5 - 3*x^4 - 2*x^3 + 2*x + 1) / ((x^2-x-1)*(x^2+x-1)). (End)
%e A189761 For n=55, the residues are {0, 1, 2, 3, 5, 8, 13, 21, 34, 47, 52, 54} which can also be written as {0, 1, 2, 3, 5, 8, 13, 21, -21, -8, -3, -1}.
%t A189761 Union[{2, 5}, Fibonacci[Range[2, 20, 2]], LucasL[Range[1, 20, 2]]]
%o A189761 (PARI) Vec(x*(x^8+x^7-x^6-2*x^5-3*x^4-2*x^3+2*x+1)/((x^2-x-1)*(x^2+x-1)) + O(x^100)) \\ _Colin Barker_, Oct 29 2013
%Y A189761 Cf. A001906, A002878, A032766, A066853, A109794, A189768.
%K A189761 nonn,easy
%O A189761 1,2
%A A189761 _T. D. Noe_, May 10 2011