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A101907
Numbers n-1 such that the arithmetic mean of the first n Fibonacci numbers (beginning with F(0)) is an integer.
6
0, 3, 5, 8, 10, 18, 23, 28, 30, 33, 40, 45, 47, 58, 60, 70, 71, 78, 88, 93, 95, 99, 100, 105, 108, 119, 128, 130, 138, 143, 148, 150, 165, 178, 180, 190, 191, 198, 200, 210, 213, 215, 219, 225, 228, 238, 239, 240, 248, 250, 268, 270, 273, 280, 287, 310, 320, 330
OFFSET
1,2
COMMENTS
The sum of the first n Fibonacci numbers is F(n+2)-1, sequence A000071.
Knott discusses the factorization of these numbers. - T. D. Noe, Oct 10 2005
LINKS
Y. Bugeaud, F. Luca, M. Mignotte and S. Siksek, On Fibonacci numbers with few prime divisors, Proc. Japan Acad., 81, Ser. A (2005), pp. 17-20.
H. R. Morton, Fibonacci-like sequences and greatest common divisors, The American Mathematical Monthly, Vol. 102, No. 8 (October 1995), pp. 731-734.
M. Ward, The prime divisors of Fibonacci numbers, Pacific J. Math., Vol. 11, No. 1 (1961), pp. 379-386.
Eric W. Weisstein's World of Mathematics, Arithmetic mean
Eric W. Weisstein's World of Mathematics, Fibonacci
FORMULA
Numbers n-1 such that (F(0)+ F(1)+ ... + F(n-1)) / n is an integer. F(i) is the i-th Fibonacci number.
a(n) = A219612(n) - 1. - Altug Alkan, Dec 29 2015
EXAMPLE
n=4 : (F(0)+F(1)+F(2)+F(3))/4 = (0+1+1+2)/4 = 1. So n-1 = 4-1 = 3 is a term.
n=6 : (F(0)+F(1)+F(2)+F(3)+F(4)+F(5))/6 = (0+1+1+2+3+5)/6 = 2. So n-1 = 6-1 = 5 is a term.
MATHEMATICA
Select[ Range[0, 500], Mod[Fibonacci[ # + 2] - 1, # + 1] == 0 &] (* Robert G. Wilson v *)
PROG
(PARI) is(n)=((Mod([1, 1; 1, 0], n+1))^(n+2))[1, 2]==1 \\ Charles R Greathouse IV, Feb 04 2013
CROSSREFS
Cf. A000045, A000071. See A111035 for another version.
Cf. A219612. - Altug Alkan, Dec 29 2015
Sequence in context: A212987 A217919 A127700 * A242250 A117668 A184410
KEYWORD
easy,nonn
AUTHOR
Ctibor O. Zizka, Jul 27 2008
EXTENSIONS
Edited and extended by Robert G. Wilson v, Aug 03 2008
Definition corrected by Altug Alkan, Dec 29 2015
STATUS
approved