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A065449
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a(n) = phi(Fibonacci(n)).
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13
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0, 1, 1, 1, 2, 4, 4, 12, 12, 16, 40, 88, 48, 232, 336, 240, 552, 1596, 1152, 4032, 3200, 5040, 17424, 28656, 12672, 60000, 120640, 89856, 188160, 514228, 288000, 1343296, 1217712, 1742400, 5697720, 6814080, 4396032, 23656320, 37691136
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OFFSET
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0,5
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COMMENTS
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For n > 4, a(n) is a multiple of 4, but a proof was elusive for a number of years. According to Koshy (2001), P. L. Montgomery "provided an elegant solution using group theory" in 1977, but Montgomery's proof is not quoted in Koshy's book.
Pe wonders if there is a closed form for this sequence, like there is for the Fibonacci numbers (Binet's formula). I wonder if there is a recurrence relation. - Alonso del Arte, Oct 11 2011
a(n) must be divisible by 4 for n > 4, since otherwise F(n) must be 1, 2, 4, a prime congruent to 3 modulo 4, or twice a prime congruent to 3 modulo 4. The first two happen for n = 1, 2, and 3, the third never occurs, the fourth can only occur for n = 4 since 3|F(4k) for all positive k, and the fifth never occurs since F(n) is never congruent to 6 modulo 8. - Charlie Neder, Apr 26 2019
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REFERENCES
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Thomas Koshy, "Fibonacci and Lucas Numbers and Applications", Wiley, New York, 2001, p. 413, Theorem 34.12.
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LINKS
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FORMULA
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a(n) >= A065451(n), with equality if and only if n = 1, 2 or 3 (Luca, 1999). - Amiram Eldar, Jan 12 2022
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EXAMPLE
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a(9) = phi(F(9)) = phi(34) = phi(2 * 17) = 16.
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MAPLE
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with(numtheory):with(combinat):a:=n->phi(fibonacci(n)): seq(a(n), n=0..38); # Zerinvary Lajos, Oct 07 2007
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MATHEMATICA
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Table[ EulerPhi[ Fibonacci[ n]], {n, 0, 46} ]
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PROG
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(PARI) for(n=1, 75, print1(eulerphi(fibonacci(n)), ", "))
(PARI) { for (n=0, 466, if (n, a=eulerphi(fibonacci(n)), a=0); write("b065449.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 20 2009
(Sage) [euler_phi(fibonacci(n))for n in range(0, 39)] # Zerinvary Lajos, Jun 06 2009
(Magma) [0] cat [EulerPhi(Fibonacci(n)): n in [1..30]]; // G. C. Greubel, Jan 18 2018
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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More terms from several correspondents, Nov 19 2001
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STATUS
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approved
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