

A065449


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


12



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
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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 3F(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
Joseph L. Pe, The Euler Phibonacci Sequence: A Problem Proposal with Software


FORMULA

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


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 xrange(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,changed


AUTHOR

Joseph L. Pe, Nov 18 2001


EXTENSIONS

More terms from several correspondents, Nov 19 2001


STATUS

approved



