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%I M0535 #57 Nov 15 2023 11:52:07
%S 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,
%T 36,50,66,28,104,30,81,60,48,72,158,36,54,72,156,40,156,42,110,152,66,
%U 46,270,78,140,96,132,52,230,120,234,108,84,58,456,60,90,228,243,144,260
%N Number of mu-atoms of period n on continent of Mandelbrot set.
%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 Vincenzo Librandi, <a href="/A006874/b006874.txt">Table of n, a(n) for n = 1..10000</a>
%H R. P. Munafo, <a href="http://www.mrob.com/pub/muency.html">Mu-Ency - The Encyclopedia of the Mandelbrot Set</a>
%H F. V. Weinstein, <a href="https://arxiv.org/abs/math/0307150">Notes on Fibonacci partitions</a>, arXiv:math/0307150 [math.NT], 2003-2018.
%F 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
%F a(1)=1; for n > 1, a(n) = Sum_{k=1..n-1} a(gcd(n,k)). - _Reinhard Zumkeller_, Sep 25 2009
%F G.f. A(x) satisfies: A(x) = x + Sum_{k>=2} phi(k) * A(x^k). - _Ilya Gutkovskiy_, Sep 04 2019
%e a(1) = 1;
%e a(2) = a(1);
%e a(3) = 2*a(1);
%e a(4) = 2*a(1) + a(2);
%e a(5) = 4*a(1);
%e a(6) = 2*a(1) + 2*a(2) + a(3);
%e a(7) = 6*a(1);
%e a(8) = 4*a(1) + 2*a(2) + a(4);
%e a(9) = 6*a(1) + 2*a(3);
%e a(10) = 4*a(1) + 4*a(2) + a(5);
%e a(11) = 10*a(1);
%e a(12) = 4*a(1) + 2*a(2) + 2*a(3) + 2*a(4) + a(6); ...
%t 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 *)
%o (PARI) a(n) = if (n==1, 1, sumdiv(n, d, if (d==1, 0, a(n/d)*eulerphi(d)))); \\ _Michel Marcus_, Apr 19 2014
%o (Python)
%o from sympy import divisors, totient
%o l=[0, 1]
%o for n in range(2, 101):
%o l.append(sum([totient(n//d)*l[d] for d in divisors(n)[:-1]]))
%o print(l[1:]) # _Indranil Ghosh_, Jul 12 2017
%o (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
%Y Cf. A000010, A006875, A006876.
%K nonn
%O 1,3
%A _Robert Munafo_, Apr 28 1994
%E More terms from _Vladeta Jovovic_, Feb 09 2002
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