A006875 Non-seed mu-atoms of period n in Mandelbrot set.

0, 1, 2, 3, 4, 7, 6, 12, 12, 23, 10, 51, 12, 75, 50, 144, 16, 324, 18, 561, 156, 1043, 22, 2340, 80, 4119, 540, 8307, 28, 17521, 30, 32928, 2096, 65567, 366, 135432, 36, 262179, 8250, 525348, 40, 1065093, 42, 2098263, 33876, 4194347, 46, 8456160, 420, 16779280

Offset: 1

Author

Robert Munafo

Keywords

• nonn

Comments

Definitions and Maxima source code on second Munafo web page. - Robert Munafo, Dec 12 2009

Examples

			From _Robert Munafo_, Dec 12 2009: (Start)
For n=1 the only mu-atom is the large cardioid, which is a seed.
For n=2 there is one, the large circular mu-atom centered at -1+0i, so a(2)=1.
For n=3 there is a seed (cardioid) at -1.75+0i, which doesn't count, and two non-seeds ("circles") at approx. -0.1225+-0.7448i, so a(3)=2. (End)
		

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

• Indranil Ghosh, Table of n, a(n) for n = 1..1000
• R. P. Munafo, Mu-Ency - The Encyclopedia of the Mandelbrot Set
• R. Munafo, Enumeration of Features [From _Robert Munafo_, Dec 12 2009]

Crossrefs

Cf. A000740, A006874, A006876, A118454.

Programs

Mathematica

Table[Sum[EulerPhi[n/d] Sum[MoebiusMu[d/c] 2^(c - 1), {c, Divisors[d]}], {d, Drop[Divisors[n], -1]}], {n, 1, 100}] (* Cheng Zhang, Apr 03 2012 *)

Python

from sympy import divisors, totient, mobius
l=[0, 0]
for n in range(2, 101):
    l.append(sum(totient(n//d)*sum(mobius(d//c)*2**(c - 1) for c in divisors(d)) for d in divisors(n)[:-1]))
print(l[1:]) # Indranil Ghosh, Jul 12 2017

Formula

a(n) = Sum_{d|n, d < n} (phi(n/d) * sum_{c|d} (mu(d/c) 2^(c-1))), where phi(n) and mu(n) are the Euler totient function (A000010) and Moebius function (A008683), respectively. - Cheng Zhang, Apr 03 2012

a(n) = A000740(n) - A006876(n).