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%I #24 Jan 04 2024 03:25:00
%S 0,3,5,8,10,18,23,28,30,33,40,45,47,58,60,70,71,78,88,93,95,99,100,
%T 105,108,119,128,130,138,143,148,150,165,178,180,190,191,198,200,210,
%U 213,215,219,225,228,238,239,240,248,250,268,270,273,280,287,310,320,330
%N Numbers n-1 such that the arithmetic mean of the first n Fibonacci numbers (beginning with F(0)) is an integer.
%C The sum of the first n Fibonacci numbers is F(n+2)-1, sequence A000071.
%C Knott discusses the factorization of these numbers. - _T. D. Noe_, Oct 10 2005
%H T. D. Noe, <a href="/A101907/b101907.txt">Table of n, a(n) for n = 1..1000</a>
%H Y. Bugeaud, F. Luca, M. Mignotte and S. Siksek, <a href="http://projecteuclid.org/euclid.pja/1116442053">On Fibonacci numbers with few prime divisors</a>, Proc. Japan Acad., 81, Ser. A (2005), pp. 17-20.
%H H. R. Morton, <a href="http://www.jstor.org/stable/2974643">Fibonacci-like sequences and greatest common divisors</a>, The American Mathematical Monthly, Vol. 102, No. 8 (October 1995), pp. 731-734.
%H M. Ward, <a href="http://projecteuclid.org/euclid.pjm/1103037560">The prime divisors of Fibonacci numbers</a>, Pacific J. Math., Vol. 11, No. 1 (1961), pp. 379-386.
%H Eric W. Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ArithmeticMean.html">Arithmetic mean</a>
%H Eric W. Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci</a>
%F Numbers n-1 such that (F(0)+ F(1)+ ... + F(n-1)) / n is an integer. F(i) is the i-th Fibonacci number.
%F a(n) = A219612(n) - 1. - _Altug Alkan_, Dec 29 2015
%e 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.
%e 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.
%t Select[ Range[0, 500], Mod[Fibonacci[ # + 2] - 1, # + 1] == 0 &] (* _Robert G. Wilson v_ *)
%o (PARI) is(n)=((Mod([1,1;1,0],n+1))^(n+2))[1,2]==1 \\ _Charles R Greathouse IV_, Feb 04 2013
%Y Cf. A000045, A000071. See A111035 for another version.
%Y Cf. A219612. - _Altug Alkan_, Dec 29 2015
%K easy,nonn
%O 1,2
%A _Ctibor O. Zizka_, Jul 27 2008
%E Edited and extended by _Robert G. Wilson v_, Aug 03 2008
%E Definition corrected by _Altug Alkan_, Dec 29 2015
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