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A006874 Number of mu-atoms of period n on continent of Mandelbrot set.
(Formerly M0535)
11
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
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.append(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
Sequence in context: A079667 A073061 A300526 * A357476 A034890 A009490
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 29 11:14 EDT 2024. Contains 371278 sequences. (Running on oeis4.)