A006875 Non-seed mu-atoms of period n in Mandelbrot set. (Formerly M0579)

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

1,3

Comments

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

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]

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).

Example

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)

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 *)

Prog

(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

Crossrefs

Cf. A000740, A006874, A006876, A118454.

Sequence in context: A106446 A321220 A036467 * A064554 A290641 A340069

Adjacent sequences: A006872 A006873 A006874 * A006876 A006877 A006878

Keyword

nonn

Author

Robert Munafo

Status

approved