A006874 Number of mu-atoms of period n on continent of Mandelbrot set.
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
Offset: 1
Keywords
Examples
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); ...
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
- Vincenzo Librandi, Table of n, a(n) for n = 1..10000
- R. P. Munafo, Mu-Ency - The Encyclopedia of the Mandelbrot Set
- F. V. Weinstein, Notes on Fibonacci partitions, arXiv:math/0307150 [math.NT], 2003-2018.
Programs
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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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Mathematica
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 *) -
PARI
a(n) = if (n==1, 1, sumdiv(n, d, if (d==1, 0, a(n/d)*eulerphi(d)))); \\ Michel Marcus, Apr 19 2014
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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]])) print(l[1:]) # Indranil Ghosh, Jul 12 2017
Formula
a(n) = Sum_{ d divides n, d1, a(1)=1, where phi is Euler totient function (A000010). - Vladeta Jovovic, Feb 09 2002
a(1)=1; for n > 1, a(n) = Sum_{k=1..n-1} a(gcd(n,k)). - Reinhard Zumkeller, Sep 25 2009
G.f. A(x) satisfies: A(x) = x + Sum_{k>=2} phi(k) * A(x^k). - Ilya Gutkovskiy, Sep 04 2019
Extensions
More terms from Vladeta Jovovic, Feb 09 2002