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A065449
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Phi(Fibonacci(n)).
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8
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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
(list;
graph;
refs;
listen;
history;
text;
internal format)
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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. [From Alonso del Arte, Oct 11 2011]
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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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Charles R Greathouse IV and Harry J. Smith, Table of n, a(n) for n = 0..1000 (Harry J. Smith to 466)
Blair Kelly, Fibonacci and Lucas Factorizations
Joseph L. Pe, The Euler Phibonacci Sequence: A Problem Proposal with Software
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FORMULA
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a(n) = A000010(A000045(n)).
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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 (zerinvarylajos(AT)yahoo.com), 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)), ", "))
(Sage) [euler_phi(fibonacci(n))for n in xrange(0, 39)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 06 2009]
(PARI) { for (n=0, 466, if (n, a=eulerphi(fibonacci(n)), a=0); write("b065449.txt", n, " ", a) ) } [From Harry J. Smith, Oct 20 2009]
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CROSSREFS
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Cf. A065451.
Sequence in context: A186973 A225232 A000936 * A130618 A129882 A129017
Adjacent sequences: A065446 A065447 A065448 * A065450 A065451 A065452
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KEYWORD
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nonn,nice
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AUTHOR
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Joseph L. Pe (joseph_l_pe(AT)hotmail.com), Nov 18 2001
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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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