A006874 Number of mu-atoms of period n on continent of Mandelbrot set.

1, 1, 2, 3, 4, 6, 6, 9, 10, 12, 10, 22, 12, 18, 24, 27, 16, 38, 18, 44, 36, 30, 22, 78, 36, 36, 50, 66, 28, 104, 30, 81, 60, 48, 72, 158, 36, 54, 72, 156, 40, 156, 42, 110, 152, 66, 46, 270, 78, 140, 96, 132, 52, 230, 120, 234, 108, 84, 58, 456, 60, 90, 228, 243, 144, 260

Offset: 1

Author

Robert Munafo, Apr 28 1994

Keywords

• nonn

Examples

			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); ...
		

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.

Crossrefs

Cf. A000010, A006875, A006876.

Programs

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

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 *)

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

Formula

a(n) = Sum_{ d divides n, d1, 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

Extensions

More terms from Vladeta Jovovic, Feb 09 2002