

A189761


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


1



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



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 evenindex Fibonacci and oddindexed 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



FORMULA

a(n) = 3*a(n2)  a(n4) 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^2x1)*(x^2+x1)). (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^7x^62*x^53*x^42*x^3+2*x+1)/((x^2x1)*(x^2+x1)) + O(x^100)) \\ Colin Barker, Oct 29 2013


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



