A101907 Numbers k such that the arithmetic mean of the first k+1 Fibonacci numbers (beginning with F(0)) is an integer.

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

Author

Ctibor O. Zizka, Jul 27 2008

Keywords

• easy
• nonn

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

Examples

			k = 3 : (F(0)+F(1)+F(2)+F(3))/4 = (0+1+1+2)/4 = 1. So 3 is a term.
k = 5 : (F(0)+F(1)+F(2)+F(3)+F(4)+F(5))/6 = (0+1+1+2+3+5)/6 = 2. So 5 is a term.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
• Yann Bugeaud, Florian Luca, Maurice Mignotte, and Samir Siksek, On Fibonacci numbers with few prime divisors, Proc. Japan Acad., Ser. A, Vol. 81, No. 2 (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.
• Morgan 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 Number.

Crossrefs

Cf. A000045, A000071. See A111035 for another version.

Cf. A219612. - Altug Alkan, Dec 29 2015

Programs

Mathematica

Select[ Range[0, 500], Mod[Fibonacci[ # + 2] - 1, # + 1] == 0 &] (* Robert G. Wilson v *)

PARI

is(n)=((Mod([1,1;1,0],n+1))^(n+2))[1,2]==1 \\ Charles R Greathouse IV, Feb 04 2013

Formula

Numbers k such that (F(0)+ F(1)+ ... + F(k)) / (k+1) is an integer, where F(i) is the i-th Fibonacci number.

a(n) = A219612(n) - 1. - Altug Alkan, Dec 29 2015

Extensions

Edited and extended by Robert G. Wilson v, Aug 03 2008

Definition corrected by Altug Alkan, Dec 29 2015