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A006874
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Number of mu-atoms of period n on continent of Mandelbrot set.
(Formerly M0535)
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11
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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
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OFFSET
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1,3
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REFERENCES
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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).
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LINKS
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FORMULA
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a(n) = Sum_{ d divides n, d<n} phi(n/d)*a(d), n>1, a(1)=1, where phi is Euler totient function (A000010). - Vladeta Jovovic, Feb 09 2002
G.f. A(x) satisfies: A(x) = x + Sum_{k>=2} phi(k) * A(x^k). - Ilya Gutkovskiy, Sep 04 2019
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EXAMPLE
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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); ...
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MATHEMATICA
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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 *)
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PROG
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(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]]))
(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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