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A006874 Mu-atoms of period n on continent of Mandelbrot set.
(Formerly M0535)
8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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.

FORMULA

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

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

EXAMPLE

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); ...

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 *)

PROG

(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+=[sum([totient(n/d)*l[d] for d in divisors(n)[:-1]]), ]

print l[1:] # Indranil Ghosh, Jul 12 2017

(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

CROSSREFS

Cf. A006875, A006876.

Sequence in context: A079667 A073061 A300526 * A034890 A009490 A243930

Adjacent sequences:  A006871 A006872 A006873 * A006875 A006876 A006877

KEYWORD

nonn

AUTHOR

Robert Munafo, Apr 28 1994

EXTENSIONS

More terms from Vladeta Jovovic, Feb 09 2002

STATUS

approved

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Last modified March 31 21:49 EDT 2020. Contains 333151 sequences. (Running on oeis4.)