cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A006874 Number of mu-atoms of period n on continent of Mandelbrot set.

Original entry on oeis.org

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

Views

Author

Robert Munafo, Apr 28 1994

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

Crossrefs

Cf. A000010, A006875, A006876.

Programs

  • 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
  • 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
  • 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

A118454 Algebraic degree of the onset of the logistic map n-bifurcation.

Original entry on oeis.org

1, 1, 2, 2, 22, 40, 114, 12, 480, 944, 2026, 3918, 8166, 16104, 32630, 240, 131038, 260928, 524250, 1046418, 2096706, 4190168, 8388562, 16768200, 33554240, 67092432, 134216136, 268402446, 536870854, 1073672968, 2147483586, 65280, 8589928346, 17179606976, 34359737478
Offset: 1

Views

Author

Eric W. Weisstein, Apr 28 2006

Keywords

Comments

a(2^n) is A087046(n).

Examples

			The onsets begin at 1, 3, 1+2*sqrt(2), 1+sqrt(6), ...

Links

Crossrefs

Cf. A000740, A006876, A086178, A086180, A087046, A091517, A098587, A118452, A118453, A118746, A181906, A181907, A181909, A181910, A181911.

Programs

  • Mathematica
    degRp[n_] := Sum[MoebiusMu[n/d] 2^(d - 1), {d, Divisors[n]}]; degRo[n_] := degRp[n]*2 - Sum[EulerPhi[n/d] degRp[d], {d, Divisors[n]}]; Table[If[n <= 2, 1, 2 If[2^Round[Log2[n]] == n, degRp[n/2], degRo[n]]], {n, 1, 35}] (* Cheng Zhang, Apr 02 2012 *)

Extensions

More terms from Cheng Zhang, Apr 02 2012

A006875 Non-seed mu-atoms of period n in Mandelbrot set.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 6, 12, 12, 23, 10, 51, 12, 75, 50, 144, 16, 324, 18, 561, 156, 1043, 22, 2340, 80, 4119, 540, 8307, 28, 17521, 30, 32928, 2096, 65567, 366, 135432, 36, 262179, 8250, 525348, 40, 1065093, 42, 2098263, 33876, 4194347, 46, 8456160, 420, 16779280
Offset: 1

Views

Author

Robert Munafo

Keywords

Comments

Definitions and Maxima source code on second Munafo web page. - Robert Munafo, Dec 12 2009

Examples

			From _Robert Munafo_, Dec 12 2009: (Start)
For n=1 the only mu-atom is the large cardioid, which is a seed.
For n=2 there is one, the large circular mu-atom centered at -1+0i, so a(2)=1.
For n=3 there is a seed (cardioid) at -1.75+0i, which doesn't count, and two non-seeds ("circles") at approx. -0.1225+-0.7448i, so a(3)=2. (End)

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

Crossrefs

Cf. A000740, A006874, A006876, A118454.

Programs

  • Mathematica
    Table[Sum[EulerPhi[n/d] Sum[MoebiusMu[d/c] 2^(c - 1), {c, Divisors[d]}], {d, Drop[Divisors[n], -1]}], {n, 1, 100}] (* Cheng Zhang, Apr 03 2012 *)
  • Python
    from sympy import divisors, totient, mobius
l=[0, 0]
for n in range(2, 101):
    l.append(sum(totient(n//d)*sum(mobius(d//c)*2**(c - 1) for c in divisors(d)) for d in divisors(n)[:-1]))
print(l[1:]) # Indranil Ghosh, Jul 12 2017

Formula

a(n) = Sum_{d|n, d < n} (phi(n/d) * sum_{c|d} (mu(d/c) 2^(c-1))), where phi(n) and mu(n) are the Euler totient function (A000010) and Moebius function (A008683), respectively. - Cheng Zhang, Apr 03 2012
a(n) = A000740(n) - A006876(n).
Showing 1-3 of 3 results.

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