OFFSET
1,3
REFERENCES
B. B. Mandelbrot, The Fractal Geometry of Nature, Freeman, NY, 1982, p. 183.
R. Penrose, The Emperor's New Mind, Penguin Books, NY, 1991, p. 138.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..10000
R. P. Munafo, Mu-Ency - The Encyclopedia of the Mandelbrot Set
F. V. Weinstein, Notes on Fibonacci partitions, arXiv:math/0307150 [math.NT], 2003-2018.
FORMULA
a(n) = Sum_{ d divides n, d<n} phi(n/d)*a(d), n>1, a(1)=1, where phi is Euler totient function (A000010). - Vladeta Jovovic, Feb 09 2002
a(1)=1; for n > 1, a(n) = Sum_{k=1..n-1} a(gcd(n,k)). - Reinhard Zumkeller, Sep 25 2009
G.f. A(x) satisfies: A(x) = x + Sum_{k>=2} phi(k) * A(x^k). - Ilya Gutkovskiy, Sep 04 2019
EXAMPLE
a(1) = 1;
a(2) = a(1);
a(3) = 2*a(1);
a(4) = 2*a(1) + a(2);
a(5) = 4*a(1);
a(6) = 2*a(1) + 2*a(2) + a(3);
a(7) = 6*a(1);
a(8) = 4*a(1) + 2*a(2) + a(4);
a(9) = 6*a(1) + 2*a(3);
a(10) = 4*a(1) + 4*a(2) + a(5);
a(11) = 10*a(1);
a(12) = 4*a(1) + 2*a(2) + 2*a(3) + 2*a(4) + a(6); ...
MATHEMATICA
a[1] = 1; a[n_] := a[n] = Block[{d = Most@Divisors@n}, Plus @@ (EulerPhi[n/d]*a /@ d)]; Array[a, 66] (* Robert G. Wilson v, Nov 22 2005 *)
PROG
(PARI) a(n) = if (n==1, 1, sumdiv(n, d, if (d==1, 0, a(n/d)*eulerphi(d)))); \\ Michel Marcus, Apr 19 2014
(Python)
from sympy import divisors, totient
l=[0, 1]
for n in range(2, 101):
l.append(sum([totient(n//d)*l[d] for d in divisors(n)[:-1]]))
print(l[1:]) # Indranil Ghosh, Jul 12 2017
(Magma) sol:=[1]; for n in [2..66] do Append(~sol, &+[sol[Gcd(n, k)]:k in [1..n-1]]); end for; sol; // Marius A. Burtea, Sep 05 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Robert Munafo, Apr 28 1994
EXTENSIONS
More terms from Vladeta Jovovic, Feb 09 2002
STATUS
approved