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

1, 2, 3, 4, 5, 8, 11, 21, 29, 55, 76, 144, 199, 377, 521, 987, 1364, 2584, 3571, 6765, 9349, 17711, 24476, 46368, 64079, 121393, 167761, 317811, 439204, 832040, 1149851, 2178309, 3010349, 5702887, 7881196, 14930352, 20633239, 39088169, 54018521, 102334155

Offset

1,2

Comments

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).

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.

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.

Links

Table of n, a(n) for n=1..40.

Index entries for linear recurrences with constant coefficients, signature (0,3,0,-1).

Formula

From Colin Barker, Oct 29 2013: (Start)

a(n) = 3*a(n-2) - a(n-4) for n > 9.

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)

Example

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}.

Mathematica

Union[{2, 5}, Fibonacci[Range[2, 20, 2]], LucasL[Range[1, 20, 2]]]

Prog

(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

Crossrefs

Cf. A001906, A002878, A032766, A066853, A109794, A189768.

Sequence in context: A135318 A374782 A210671 * A329576 A101137 A256386

Adjacent sequences: A189758 A189759 A189760 * A189762 A189763 A189764

Keyword

nonn,easy

Author

T. D. Noe, May 10 2011

Status

approved