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%I #58 Feb 17 2022 00:06:17
%S 0,1,1,1,2,4,4,12,12,16,40,88,48,232,336,240,552,1596,1152,4032,3200,
%T 5040,17424,28656,12672,60000,120640,89856,188160,514228,288000,
%U 1343296,1217712,1742400,5697720,6814080,4396032,23656320,37691136
%N a(n) = phi(Fibonacci(n)).
%C 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.
%C 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
%C 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
%D Thomas Koshy, "Fibonacci and Lucas Numbers and Applications", Wiley, New York, 2001, p. 413, Theorem 34.12.
%H Amiram Eldar, <a href="/A065449/b065449.txt">Table of n, a(n) for n = 0..1408</a> (terms 0..466 from Harry J. Smith, terms 467..1000 from Charles R Greathouse IV)
%H Blair Kelly, <a href="http://mersennus.net/fibonacci/">Fibonacci and Lucas Factorizations</a>.
%H Florian Luca, <a href="https://www.fq.math.ca/Scanned/37-3/luca.pdf">Arithmetic Functions of Fibonacci Numbers</a>, The Fibonacci Quarterly, Vol. 37, No. 3 (1999), pp. 265-268.
%H Joseph L. Pe, <a href="http://numeratus.net/phibonacci/phibonacci.html">The Euler Phibonacci Sequence: A Problem Proposal with Software</a>, 2001.
%F a(n) = A000010(A000045(n)).
%F a(n) >= A065451(n), with equality if and only if n = 1, 2 or 3 (Luca, 1999). - _Amiram Eldar_, Jan 12 2022
%e a(9) = phi(F(9)) = phi(34) = phi(2 * 17) = 16.
%p with(numtheory):with(combinat):a:=n->phi(fibonacci(n)): seq(a(n), n=0..38); # _Zerinvary Lajos_, Oct 07 2007
%t Table[ EulerPhi[ Fibonacci[ n]], {n, 0, 46} ]
%o (PARI) for(n=1,75,print1(eulerphi(fibonacci(n)),","))
%o (PARI) { for (n=0, 466, if (n, a=eulerphi(fibonacci(n)), a=0); write("b065449.txt", n, " ", a) ) } \\ _Harry J. Smith_, Oct 20 2009
%o (Sage) [euler_phi(fibonacci(n))for n in range(0,39)] # _Zerinvary Lajos_, Jun 06 2009
%o (Magma) [0] cat [EulerPhi(Fibonacci(n)): n in [1..30]]; // _G. C. Greubel_, Jan 18 2018
%Y Cf. A000010, A000045, A065451.
%K nonn,nice
%O 0,5
%A _Joseph L. Pe_, Nov 18 2001
%E More terms from several correspondents, Nov 19 2001
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