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A104714
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Greatest common divisor of a Fibonacci number and its index.
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9
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0, 1, 1, 1, 1, 5, 2, 1, 1, 1, 5, 1, 12, 1, 1, 5, 1, 1, 2, 1, 5, 1, 1, 1, 24, 25, 1, 1, 1, 1, 10, 1, 1, 1, 1, 5, 36, 1, 1, 1, 5, 1, 2, 1, 1, 5, 1, 1, 48, 1, 25, 1, 1, 1, 2, 5, 7, 1, 1, 1, 60, 1, 1, 1, 1, 5, 2, 1, 1, 1, 5, 1, 72, 1, 1, 25, 1, 1, 2, 1, 5, 1, 1, 1, 12, 5, 1, 1, 1, 1, 10, 13, 1, 1, 1, 5, 96, 1
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OFFSET
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0,6
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COMMENTS
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Considering this sequence is a natural sequel to the investigation of the problem when F_n is divisible by n (the numbers occurring in A023172). This sequence has several nice properties. (1) n | m implies a(n) | a(m) for arbitrary naturals n and m. This property is a direct consequence of the analogous well-known property of Fibonacci numbers. (2) gcd (a(n), a(m)) = a(gcd(n, m)) for arbitrary naturals n and m. Also this property follows directly from the analogous (perhaps not so well-known) property of Fibonacci numbers. (3) a(n) * a(m) | a(n * m) for arbitrary naturals n and m. This property is remarkable especially in the light that the analogous proposition for Fibonacci numbers fails if n and m are not relatively prime (e.g. F_3 * F_3 does not divide F_9). (4) The set of numbers satisfying a(n) = n is closed w.r.t. multiplication. This follows easily from (3).
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..20000 (first 1001 terms from T. D. Noe)
Paolo Leonetti, Carlo Sanna, On the greatest common divisor of n and the nth Fibonacci number, arXiv:1704.00151 [math.NT], 2017.
Carlo Sanna, Emanuele Tron, The density of numbers n having a prescribed G.C.D. with the nth Fibonacci number, arXiv:1705.01805 [math.NT], 2017.
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FORMULA
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a(n) = gcd (F_n, n).
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EXAMPLE
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The natural numbers: 0 1 2 3 4 5 6 7 8 9 10 11 12 ...
The Fibonacci numbers: 0 1 1 2 3 5 8 13 21 34 55 89 144 ...
The corresponding GCDs: 0 1 1 1 1 5 2 1 1 1 5 1 12 ...
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MAPLE
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b:= proc(n) option remember; local r, M, p; r, M, p:=
<<1|0>, <0|1>>, <<0|1>, <1|1>>, n;
do if irem(p, 2, 'p')=1 then r:= r.M mod n fi;
if p=0 then break fi; M:= M.M mod n
od; r[1, 2]
end:
a:= n-> igcd(n, b(n)):
seq(a(n), n=0..100); # Alois P. Heinz, Apr 05 2017
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MATHEMATICA
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Table[GCD[Fibonacci[n], n], {n, 0, 97}] (* Alonso del Arte, Nov 22 2010 *)
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PROG
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(Haskell) let fibs@(_ : fs) = 0 : 1 : zipWith (+) fibs fs in 0 : zipWith gcd [1 ..] fs
(PARI) a(n)=if(n, gcd(n, lift(Mod([1, 1; 1, 0], n)^n)[1, 2]), 0) \\ Charles R Greathouse IV, Sep 24 2013
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CROSSREFS
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Cf. A023172, A000045, A001177, A001175, A001176. a(n) = gcd(A000045(n), A001477(n)). a(n) = n iff n occurs in A023172 iff n | A000045(n).
Cf. A074215 (a(n)==1).
Sequence in context: A033325 A126690 A263007 * A085119 A010128 A180133
Adjacent sequences: A104711 A104712 A104713 * A104715 A104716 A104717
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KEYWORD
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easy,nonn
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AUTHOR
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Harmel Nestra (harmel.nestra(AT)ut.ee), Apr 23 2005
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STATUS
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approved
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