A065449 -id:A065449 - cp's OEIS frontend

A272040 a(n) = A000010(A000129(n)).

1, 1, 4, 4, 28, 24, 156, 128, 784, 1120, 5740, 2880, 33460, 37128, 150080, 147456, 1128256, 931392, 6446016, 4677120, 28514304, 44450560, 224075664, 106168320, 1265644800, 1560708240, 5970392064, 5588803584, 44560482148, 33497856000, 255263424000, 196368924672, 1210784762880

Offset: 1

Altug Alkan, May 06 2016

nonn

			a(3) = 4 because a(3) = A000010(A000129(3)) = A000010(5) = 4.

Amiram Eldar, Table of n, a(n) for n = 1..630

Cf. A000010, A000129, A000203, A065449, A363829, A363831, A363833, A364818.

EulerPhi[LinearRecurrence[{2, 1}, {1, 2}, 33]] (* Amiram Eldar, Oct 21 2023 *)

a000129(n) = ([2,1;1,0]^n)[2,1];
a(n) = eulerphi(a000129(n));

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)).

A065451 a(n) = Fibonacci(phi(n)), a(0) = 0.

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 8, 3, 8, 3, 55, 3, 144, 8, 21, 21, 987, 8, 2584, 21, 144, 55, 17711, 21, 6765, 144, 2584, 144, 317811, 21, 832040, 987, 6765, 987, 46368, 144, 14930352, 2584, 46368, 987, 102334155, 144, 267914296, 6765, 46368, 17711, 1836311903

Offset: 0

Joseph L. Pe, Nov 18 2001

nonn

			a(13) = F(phi(13)) = F(12) = 144.

Harry J. Smith, Table of n, a(n) for n = 0..1000
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, A065449 (phi(Fibonacci(n))).

[0] cat [Fibonacci(EulerPhi(n)): n in [1..50]]; // G. C. Greubel, Jan 18 2018

Table[ Fibonacci[ EulerPhi[ n]], {n, 0, 60} ]

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

{ for (n=0, 1000, if (n, a=fibonacci(eulerphi(n)), a=0); write("b065451.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 20 2009

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

a(n) <= A065449(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

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.

A075775 Numbers k that divide phi(Fibonacci(k)).

Original entry on oeis.org

1, 10, 11, 12, 14, 15, 18, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 88, 89, 90, 91, 92, 93

Offset: 1

Benoit Cloitre, Oct 09 2002

This sequence is infinite (Luca, 2002). - Amiram Eldar, Jan 12 2022

Amiram Eldar, Table of n, a(n) for n = 1..1343
Florian Luca, Problem H-590, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 40, No. 5 (2002), p. 472; Arithmetic Functions of Fibonacci Numbers, Solution to Problem H-590 by J.-Ch. Schlage-Puchta and J. Spilker, ibid., Vol. 41, No. 4 (2002), pp. 382-384.

Cf. A000010, A000045, A065449.

Select[Range[100],Divisible[EulerPhi[Fibonacci[#]],#]&] (* Harvey P. Dale, Jun 11 2015 *)

A181056 a(n) = prime(phi(Fibonacci(n))).

Original entry on oeis.org

2, 2, 2, 3, 7, 7, 37, 37, 53, 173, 457, 223, 1459, 2267, 1511, 4003, 13463, 9311, 38197, 29443, 49033, 193093, 333227, 136069, 746773, 1592923, 1157579, 2575043, 7594759, 4073233, 21225769, 19112567, 28016189, 98825561, 119488379, 75032131, 446083661, 729322973

Offset: 1

Carmine Suriano, Oct 01 2010

nonn

Phi is Euler's totient function A000010.

			a(7) = 37 since prime(phi(fib(7))) = prime(phi(13)) = prime(12) = 37 is the 12th prime.

Amiram Eldar, Table of n, a(n) for n = 1..85 (terms 1..60 from Vincenzo Librandi)

Cf. A000010, A000040, A000045, A065449, A181058.

[NthPrime(EulerPhi(Fibonacci(n))): n in [1..40]]; // Vincenzo Librandi, Jun 23 2014

A065449 := proc(n) numtheory[phi](combinat[fibonacci](n)) ; end proc: A181056 := proc(n) ithprime(A065449(n)) ; end proc: seq(A181056(n),n=1..32) ; # R. J. Mathar, Oct 02 2010

Prime[EulerPhi[Fibonacci[Range[35]]]] (* Harvey P. Dale, Jun 22 2014 *)

a(n) = A000040(A065449(n)). - R. J. Mathar, Oct 02 2010

a(1) inserted by R. J. Mathar, Oct 02 2010

A075776 Numbers k that do not divide phi(Fibonacci(k)).

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 13, 16, 17, 19, 23, 31, 32, 43, 47, 57, 59, 61, 71, 79, 83, 87, 101, 109, 129, 137, 151, 161, 183, 187, 191, 199, 213, 221, 247, 251, 271, 311, 347, 381, 391, 393, 433, 493, 497, 499, 599, 749, 767, 827, 839, 913, 943, 983, 991, 1017, 1027

Offset: 1

Benoit Cloitre, Oct 09 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..65

Cf. A000010, A000045, A065449.

Select[Range[200], !Divisible[EulerPhi[Fibonacci[#]], #] &] (* Amiram Eldar, Jun 26 2022 *)

isok(n) = eulerphi(fibonacci(n)) % n; \\ Michel Marcus, Nov 29 2013

More terms from Michel Marcus, Nov 29 2013

a(43)-a(59) from Amiram Eldar, Jun 26 2022

A107647 Euler's totient function applied to tribonacci numbers.

Original entry on oeis.org

1, 1, 1, 2, 6, 12, 8, 20, 54, 148, 136, 144, 612, 1200, 1344, 2448, 10506, 16848, 13824, 22000, 83232, 148716, 205368, 377736, 920160, 1694088, 1880304, 3290112, 14839968, 22472640, 17805312, 42407136, 117876096, 327661128, 178588800, 561863168, 1604383200

Offset: 2

Roger L. Bagula, Jun 09 2005

nonn

Amiram Eldar, Table of n, a(n) for n = 2..365

Cf. A000010, A000073, A065449, A197218, A272040.

F[1] = 0; F[2] = 1; F[3] = 1; F[n__] := F[n] = F[n - 1] + F[n - 2] + F[n - 3]; Table[EulerPhi[F[n]], {n, 2, 50}]

a(n) = A000010(A000073(n)). - Amiram Eldar, Mar 02 2020

First term 0 removed and offset corrected by Amiram Eldar, Mar 02 2020

A277261 Least k>0 such that phi(Fibonacci(n)) divides phi(Fibonacci(n+k)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 4, 3, 5, 11, 2, 13, 5, 6, 16, 17, 6, 16, 10, 14, 11, 23, 20, 25, 13, 22, 14, 29, 10, 31, 32, 22, 17, 35, 36, 37, 38, 26, 20, 41, 42, 43, 44, 18, 46, 47, 48, 49, 50, 51, 26, 53, 54, 55, 56, 57, 58, 59, 60, 61, 31, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 37, 75, 76, 77, 78, 79, 80, 81, 41

Offset: 1

Altug Alkan, Oct 07 2016

nonn

a(n) <= n, since Fibonacci(n) divides Fibonacci(2n) and phi(x) divides phi(y) if x divides y. - Robert Israel, Dec 01 2016

			a(7) = 1 because phi(Fibonacci(7)) = phi(Fibonacci(8)) = 12.

Cf. A000010, A000045, A065449.

f:= proc(n) uses combinat, numtheory; local k, phin;
    phin:= phi(fibonacci(n));
    for k from 1 do if phi(fibonacci(n+k)) mod phin = 0 then return k fi od
end proc;
map(f, [$1..100]); # Robert Israel, Dec 01 2016

Table[k = 1; While[Mod[EulerPhi@ Fibonacci[n + k], EulerPhi@ Fibonacci@ n] != 0, k++]; k, {n, 82}] (* Michael De Vlieger, Nov 23 2016 *)

a(n) = {my(k=1); while (eulerphi(fibonacci(n+k)) % eulerphi(fibonacci(n)), k++); k;} \\ Michel Marcus, Nov 19 2016

A366773 a(n) = A000010(A001045(n)).

Original entry on oeis.org

1, 1, 2, 4, 10, 12, 42, 64, 108, 300, 682, 576, 2730, 5292, 6600, 16384, 43690, 46656, 174762, 240000, 455112, 1320352, 2796202, 2211840, 10125000, 22358700, 28256040, 66382848, 175923744, 178200000, 715827882, 1073741824, 1877540544, 5726448300, 10133592000

Offset: 1

Sean A. Irvine, Oct 21 2023

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1122

Cf. A000010, A001045, A065449, A197218, A272040.

a(n) = phi(A001045(n)).

cp's OEIS Frontend

A272040 a(n) = A000010(A000129(n)).

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

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A065451 a(n) = Fibonacci(phi(n)), a(0) = 0.

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

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A075775 Numbers k that divide phi(Fibonacci(k)).

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A181056 a(n) = prime(phi(Fibonacci(n))).

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A075776 Numbers k that do not divide phi(Fibonacci(k)).

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