

A101907


Numbers n1 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
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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

T. D. Noe, Table of n, a(n) for n = 1..1000
Y. Bugeaud, F. Luca, M. Mignotte and S. Siksek, On Fibonacci numbers with few prime divisors, Proc. Japan Acad., 81, Ser. A (2005), pp. 1720. [From Ctibor O. Zizka, Aug 06 2008]
H. R. Morton, Fibonaccilike sequences and greatest common divisors, The American Mathematical Monthly, Vol. 102, No. 8 (October 1995), pp. 731734 . [From Ctibor O. Zizka, Aug 06 2008]
M. Ward, The prime divisors of Fibonacci numbers, Pacific J. Math., Vol. 11, No. 1 (1961), pp. 379386. [From Ctibor O. Zizka, Aug 06 2008]
Eric W. Weisstein's World of Mathematics, Arithmetic mean
Eric W. Weisstein's World of Mathematics, Fibonacci


FORMULA

Numbers n1 such that (F(0)+ F(1)+ ... + F(n1)) / n is an integer. F(i) is the ith 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 n1 = 41 = 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 n1 = 61 = 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
Adjacent sequences: A101904 A101905 A101906 * A101908 A101909 A101910


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



