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

Author

Joseph L. Pe, Nov 18 2001

Keywords

• nonn
• nice

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

Examples

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

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.

Crossrefs

Cf. A000010, A000045, A065451.

Programs

Magma

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

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} ]

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

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

Extensions

More terms from several correspondents, Nov 19 2001