A065451 -id:A065451 - cp's OEIS frontend

A065449 a(n) = phi(Fibonacci(n)).

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

Offset: 0

Joseph L. Pe, Nov 18 2001

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

			a(9) = phi(F(9)) = phi(34) = phi(2 * 17) = 16.

Thomas Koshy, "Fibonacci and Lucas Numbers and Applications", Wiley, New York, 2001, p. 413, Theorem 34.12.

Amiram Eldar, Table of n, a(n) for n = 0..1408 (terms 0..466 from Harry J. Smith, terms 467..1000 from Charles R Greathouse IV)
Blair Kelly, Fibonacci and Lucas Factorizations.
Florian Luca, Arithmetic Functions of Fibonacci Numbers, The Fibonacci Quarterly, Vol. 37, No. 3 (1999), pp. 265-268.
Joseph L. Pe, The Euler Phibonacci Sequence: A Problem Proposal with Software, 2001.

Cf. A000010, A000045, A065451.

[0] cat [EulerPhi(Fibonacci(n)): n in [1..30]]; // G. C. Greubel, Jan 18 2018

with(numtheory):with(combinat):a:=n->phi(fibonacci(n)): seq(a(n), n=0..38); # Zerinvary Lajos, Oct 07 2007

Table[ EulerPhi[ Fibonacci[ n]], {n, 0, 46} ]

for(n=1,75,print1(eulerphi(fibonacci(n)),","))

{ for (n=0, 466, if (n, a=eulerphi(fibonacci(n)), a=0); write("b065449.txt", n, " ", a) ) } \\  Harry J. Smith, Oct 20 2009

[euler_phi(fibonacci(n))for n in range(0,39)] # Zerinvary Lajos, Jun 06 2009

a(n) = A000010(A000045(n)).

a(n) >= A065451(n), with equality if and only if n = 1, 2 or 3 (Luca, 1999). - Amiram Eldar, Jan 12 2022

More terms from several correspondents, Nov 19 2001

A197218 a(n) = phi(Lucas(n)).

Original entry on oeis.org

1, 1, 2, 2, 6, 10, 6, 28, 46, 36, 80, 198, 132, 520, 560, 600, 2206, 3570, 1908, 9348, 12960, 11760, 25704, 63480, 50692, 150000, 180960, 208008, 609084, 1130304, 604800, 3010348, 4865280, 3920400, 8374344, 17836000, 13685760, 54018520, 58269200, 69600960

Offset: 0

T. D. Noe, Oct 12 2011

nonn

Amiram Eldar, Table of n, a(n) for n = 0..1000 (using Blair Kelly's data; terms 0..200 from Vincenzo Librandi)
Blair Kelly, Fibonacci and Lucas Factorizations

Cf. A000010, A000032, A065449, A065451, A197219 (Lucas(phi(n))).

[EulerPhi(Lucas(n)): n in [0..40]]; // Vincenzo Librandi, Oct 13 2011

Mathematica
```
Table[EulerPhi[LucasL[n]], {n, 0, 40}]
```

for(n=0,30, print1(eulerphi(fibonacci(n+1) + fibonacci(n-1)), ", ")) \\ G. C. Greubel, Dec 22 2017

a(n) = A000010(A000032(n)).

A197219 a(0) = 2, a(n) = Lucas(phi(n)) for n > 0.

Original entry on oeis.org

2, 1, 1, 3, 3, 7, 3, 18, 7, 18, 7, 123, 7, 322, 18, 47, 47, 2207, 18, 5778, 47, 322, 123, 39603, 47, 15127, 322, 5778, 322, 710647, 47, 1860498, 2207, 15127, 2207, 103682, 322, 33385282, 5778, 103682, 2207, 228826127, 322, 599074578, 15127, 103682, 39603

Offset: 0

T. D. Noe, Oct 12 2011

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000010, A000032, A065449, A065451, A197218 (phi(Lucas(n))).

[2] cat [Lucas(EulerPhi(n)): n in [1..60]]; // Vincenzo Librandi, Oct 13 2011

Mathematica
```
Table[LucasL[EulerPhi[n]], {n, 0, 50}]
```

a(n) = if(n==0, 2, fibonacci(eulerphi(n)+1) + fibonacci(eulerphi(n)-1)) \\ G. C. Greubel, Dec 22 2017

a(n) = A000032(A000010(n)) for n > 0.

A181058 a(n) = prime(Fibonacci(phi(n))), where prime = A000040, Fibonacci = A000045 and phi = A000010.

Original entry on oeis.org

2, 2, 2, 2, 5, 2, 19, 5, 19, 5, 257, 5, 827, 19, 73, 73, 7793, 19, 23159, 73, 827, 257, 196687, 73, 67931, 827, 23159, 827, 4528949, 73, 12717703, 7793, 67931, 7793, 563987, 827, 274253209, 23159, 563987, 7793, 2088145739, 827, 5738374519, 67931

Offset: 1

Carmine Suriano, Oct 01 2010

nonn

Phi is Euler's totient function A000010.

			a(7) = 19 since prime(fib(phi(7))) = prime(fib(6)) = prime(8) = 19 that is the 8th prime.

Amiram Eldar, Table of n, a(n) for n = 1..88

Cf. A000010, A000040, A000045, A030427, A065451, A181056.

f[n_] := Prime@ Fibonacci@ EulerPhi@ n; Array[f, 44] (* Robert G. Wilson v, Oct 02 2010 *)

A181058(n) = prime(fibonacci(eulerphi(n))); \\ Antti Karttunen, Dec 06 2017

a(n) = A000040(A065451(n)) = A030427(A000010(n)). - Antti Karttunen, Dec 06 2017

More terms from Robert G. Wilson v, Oct 02 2010

cp's OEIS Frontend

A065449 a(n) = phi(Fibonacci(n)).

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A197218 a(n) = phi(Lucas(n)).

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A197219 a(0) = 2, a(n) = Lucas(phi(n)) for n > 0.

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A181058 a(n) = prime(Fibonacci(phi(n))), where prime = A000040, Fibonacci = A000045 and phi = A000010.

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