

A006875


Nonseed muatoms of period n in Mandelbrot set.
(Formerly M0579)


3



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
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OFFSET

1,3


COMMENTS

Definitions and Maxima source code on second Munafo web page. [From 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, MuEncy  The Encyclopedia of the Mandelbrot Set
R. Munafo, Enumeration of Features [From Robert Munafo, Dec 12 2009]


FORMULA

a(n) = sum_{dn, d < n} (phi(n/d) * sum_{cd} (mu(d/c) 2^(c1))), 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

Contribution from Robert Munafo, Dec 12 2009: (Start)
For n=1 the only muatom is the large cardioid, which is a seed.
For n=2 there is one, the large circular muatom 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 nonseeds ("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 xrange(2, 101):l+=[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: A091204 A106446 A036467 * A064554 A290641 A265352
Adjacent sequences: A006872 A006873 A006874 * A006876 A006877 A006878


KEYWORD

nonn


AUTHOR

Robert Munafo


STATUS

approved



