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

0,5

Comments

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

References

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

Links

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.

Formula

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

Example

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

Maple

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A000010, A000045, A065451.

Sequence in context: A292303 A000936 A319594 * A130618 A129882 A129017

Adjacent sequences: A065446 A065447 A065448 * A065450 A065451 A065452

Keyword

nonn,nice

Author

Joseph L. Pe, Nov 18 2001

Extensions

More terms from several correspondents, Nov 19 2001

Status

approved