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%I M0579 #37 Jan 25 2022 00:44:04
%S 0,1,2,3,4,7,6,12,12,23,10,51,12,75,50,144,16,324,18,561,156,1043,22,
%T 2340,80,4119,540,8307,28,17521,30,32928,2096,65567,366,135432,36,
%U 262179,8250,525348,40,1065093,42,2098263,33876,4194347,46,8456160,420,16779280
%N Non-seed mu-atoms of period n in Mandelbrot set.
%C Definitions and Maxima source code on second Munafo web page. - _Robert Munafo_, Dec 12 2009
%D B. B. Mandelbrot, The Fractal Geometry of Nature, Freeman, NY, 1982, p. 183.
%D R. Penrose, The Emperor's New Mind, Penguin Books, NY, 1991, p. 138.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Indranil Ghosh, <a href="/A006875/b006875.txt">Table of n, a(n) for n = 1..1000</a>
%H R. P. Munafo, <a href="http://www.mrob.com/pub/muency.html">Mu-Ency - The Encyclopedia of the Mandelbrot Set</a>
%H R. Munafo, <a href="http://www.mrob.com/pub/muency/enumerationoffeatures.html">Enumeration of Features</a> [From _Robert Munafo_, Dec 12 2009]
%F 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
%F a(n) = A000740(n) - A006876(n).
%e From _Robert Munafo_, Dec 12 2009: (Start)
%e For n=1 the only mu-atom is the large cardioid, which is a seed.
%e For n=2 there is one, the large circular mu-atom centered at -1+0i, so a(2)=1.
%e 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)
%t 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 *)
%o (Python)
%o from sympy import divisors, totient, mobius
%o l=[0, 0]
%o for n in range(2, 101):
%o l.append(sum(totient(n//d)*sum(mobius(d//c)*2**(c - 1) for c in divisors(d)) for d in divisors(n)[:-1]))
%o print(l[1:]) # _Indranil Ghosh_, Jul 12 2017
%Y Cf. A000740, A006874, A006876, A118454.
%K nonn
%O 1,3
%A _Robert Munafo_
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