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Search: id:A101907
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| A101907 |
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Numbers n such that the arithmetic mean of the first n Fibonacci numbers (beginning with F(0)) is an integer. |
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3
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| 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
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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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 (noe(AT)sspectra.com), Oct 10 2005
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REFERENCES
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H. R. Morton : Fibonacci-like sequences and greatest common divisors,The American Mathematical Monthly, Vol. 102, No. 8 (October 1995), pp. 731-734 . [From Ctibor O. Zizka (ctibor.zizka(AT)seznam.cz), Aug 06 2008]
M. Ward : The prime divisors of Fibonacci numbers, Pacific J. Math., Vol. 11, No. 1 (1961), pp. 379-386. [From Ctibor O. Zizka (ctibor.zizka(AT)seznam.cz), Aug 06 2008]
Y. Bugeaud, F. Luca, M. Mignotte and S. Siksek : On Fibonacci numbers with few prime divisors, Proc. Japan Acad. Ser. A, Math. Sci., Vol. 81, No.2 (2005), pp.17-20. [From Ctibor O. Zizka (ctibor.zizka(AT)seznam.cz), Aug 06 2008]
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
Mathworld, Arithmetic mean
Mathworld, Fibonacci
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FORMULA
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Numbers n such that (F(0)+ F(1)+ ... + F(n-1)) / n is an integer. F(i) is the i-th Fibonacci number.
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EXAMPLE
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n=4 : (F(0)+F(1)+F(2)+F(3))/4 = (0+1+1+2)/4 = 1.
n=6 : (F(0)+F(1)+F(2)+F(3)+F(4)+F(5))/6 = (0+1+1+2+3+5)/6 = 2.
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MATHEMATICA
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Select[ Range[0, 500], Mod[Fibonacci[ # + 2] - 1, # + 1] == 0 &] (* RGWv *)
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CROSSREFS
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Cf. A000045, A000071. See A111035 for another version.
Sequence in context: A133097 A117176 A127700 this_sequence A117668 A050098 A025512
Adjacent sequences: A101904 A101905 A101906 this_sequence A101908 A101909 A101910
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KEYWORD
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easy,nonn
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AUTHOR
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Ctibor O. Zizka (ctibor.zizka(AT)seznam.cz), Jul 27 2008
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EXTENSIONS
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Edited and extended by Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 3 2008
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