A006874 -id:A006874 - cp's OEIS frontend

A006876 Mu-molecules in Mandelbrot set whose seeds have period n.

1, 0, 1, 3, 11, 20, 57, 108, 240, 472, 1013, 1959, 4083, 8052, 16315, 32496, 65519, 130464, 262125, 523209, 1048353, 2095084, 4194281, 8384100, 16777120, 33546216, 67108068, 134201223, 268435427, 536836484, 1073741793, 2147417952, 4294964173, 8589803488, 17179868739

Offset: 1

Robert Munafo

nonn

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).

Cheng Zhang, Table of n, a(n) for n = 1..1000
R. P. Munafo, Enumeration of Features

Cf. A006874, A006875, A000740, A118454.

degRp[n_] := Sum[MoebiusMu[n/d] 2^(d - 1), {d, Divisors[n]}]; Table[degRp[n]*2 - Sum[EulerPhi[n/d] degRp[d], {d, Divisors[n]}], {n, 1, 100}] (* Cheng Zhang, Apr 02 2012 *)

A000740(n)=sumdiv(n,d,moebius(n/d)<<(d-1))
a(n)=2*A000740(n)-sumdiv(n, d, eulerphi(n/d)*A000740(d)) \\ Charles R Greathouse IV, Feb 18 2013

a(n) = 2*l(n) - sum_{d|n} phi(n/d)*l(d), where l(n) = sum_{d|n} mu(n/d) 2^(d-1) (A000740), and phi(n) and mu(n) are the Euler totient function (A000010) and Moebius function (A008683), respectively. - Cheng Zhang, Apr 02 2012

Web link changed to more relevant page by Robert Munafo, Nov 16 2010

More terms from Cheng Zhang, Apr 02 2012

A332791 a(1) = 1; a(n+1) = Sum_{d|n} phi(d) * a(d).

Original entry on oeis.org

1, 1, 2, 5, 12, 49, 104, 625, 2512, 15077, 60358, 603581, 2414438, 28973257, 173840168, 1390721397, 11125773688, 178012379009, 1068074289230, 19225337206141, 153802697709496, 1845632372514581, 18456323725749392, 406039121966486625, 3248312975734309938

Offset: 1

Ilya Gutkovskiy, Feb 24 2020

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..495

Cf. A000010, A006874, A038045, A057660, A307793, A307794, A332792.

a[1] = 1; a[n_] := Sum[EulerPhi[d] a[d], {d, Divisors[n - 1]}]; Table[a[n], {n, 1, 25}]
a[1] = 1; a[n_] := a[n] = Sum[a[(n - 1)/GCD[n - 1, k]], {k, 1, n - 1}]; Table[a[n], {n, 1, 25}]

a(1) = 1; a(n+1) = Sum_{k=1..n} a(n/gcd(n, k)).

a(n) = Sum_{k=1..n} a(gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021

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

Robert Munafo

nonn

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

			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)

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).

Indranil Ghosh, Table of n, a(n) for n = 1..1000
R. P. Munafo, Mu-Ency - The Encyclopedia of the Mandelbrot Set
R. Munafo, Enumeration of Features [From _Robert Munafo_, Dec 12 2009]

Cf. A000740, A006874, A006876, A118454.

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 *)

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

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).

A332792 a(1) = 1; a(n) = Sum_{d|n, d < n} phi(d) * a(d).

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 1, 6, 3, 6, 1, 16, 1, 8, 7, 30, 1, 30, 1, 34, 9, 12, 1, 104, 5, 14, 21, 60, 1, 96, 1, 270, 13, 18, 11, 278, 1, 20, 15, 330, 1, 174, 1, 136, 81, 24, 1, 1176, 7, 130, 19, 186, 1, 588, 15, 804, 21, 30, 1, 1204, 1, 32, 135, 4590, 17, 402, 1, 310, 25, 348

Offset: 1

Ilya Gutkovskiy, Feb 24 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000010, A006874, A008578 (positions of 1's), A038045, A057660, A332791.

a[1] = 1; a[n_] := Sum[If[d < n, EulerPhi[d] a[d], 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 70}]
a[1] = 1; a[n_] := a[n] = Sum[If[GCD[n, k] > 1, a[n/GCD[n, k]], 0], {k, 1, n}]; Table[a[n], {n, 1, 70}]

up_to = 20000;
A332792list(n) = { my(v=vector(n)); v[1] = 1; for(n=2, #v, v[n] = sumdiv(n, d, if(d==n, 0, v[d]*eulerphi(d)))); (v); };
v332792 = A332792list(up_to);
A332792(n) = v332792[n]; \\ Antti Karttunen, Jan 22 2025

a(1) = 1; a(n) = Sum_{k=1..n, gcd(n, k) > 1} a(n/gcd(n, k)).

A333613 a(1) = 1; thereafter a(n) = Sum_{k = 1..n} a(k/gcd(n,k)).

Original entry on oeis.org

1, 2, 4, 7, 15, 21, 51, 78, 158, 230, 568, 661, 1797, 2595, 5117, 7789, 19095, 21702, 59892, 81801, 171329, 258028, 630942, 713093, 1887828, 2776798, 5727675, 8335692, 20702970, 21420664, 62826604, 92041835, 189376593, 281410640, 656577018, 742729123, 2087788417

Offset: 1

Ilya Gutkovskiy, Mar 28 2020

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..3930
Vaclav Kotesovec, Plot of a(n)^(1/n) for n = 1..10000

Cf. A006874, A045545, A057661, A332791.

A333613:= proc(n) option remember;
if n<3 then n;
else add( A333613(lcm(n,j)/n), j = 1..n);
end if; end proc;
seq(A333613(n), n=1..40); # G. C. Greubel, Mar 08 2021

a[1] = 1; a[n_] := a[n] = Sum[a[k/GCD[n, k]], {k, n}]; Table[a[n], {n, 37}]
a[1] = 1; a[n_] := a[n] = Sum[Sum[If[GCD[k, d] == 1, a[k], 0], {k, d}], {d, Divisors[n]}]; Table[a[n], {n, 37}]

@CachedFunction
def A333613(n): return 1 if n==1 else sum( A333613(lcm(n, j)/n) for j in (1..n) )
[A333613(n) for n in (1..40)] # G. C. Greubel, Mar 08 2021

a(1) = 1; a(n) = Sum_{k = 1..n} a(lcm(n, k)/n).

a(1) = 1; a(n) = Sum_{d|n} Sum_{k = 1..d, gcd(d, k) = 1} a(k).

A338750 a(n) = 1 + Sum_{k=1..n-1} a(gcd(n,k)).

Original entry on oeis.org

1, 2, 3, 5, 5, 10, 7, 14, 13, 18, 11, 35, 13, 26, 31, 41, 17, 58, 19, 65, 45, 42, 23, 122, 41, 50, 63, 95, 29, 154, 31, 122, 73, 66, 83, 241, 37, 74, 87, 230, 41, 226, 43, 155, 193, 90, 47, 419, 85, 194, 115, 185, 53, 338, 135, 338, 129, 114, 59, 679, 61, 122, 283

Offset: 1

Ilya Gutkovskiy, Nov 06 2020

nonn

Inverse Moebius transform of A006874.

Cf. A000010, A006874, A038045, A130054.

a[n_] := a[n] = 1 + Sum[a[GCD[n, k]], {k, 1, n - 1}]; Table[a[n], {n, 1, 63}]
a[n_] := a[n] = 1 + DivisorSum[n, EulerPhi[n/#] a[#] &, # < n &]; Table[a[n], {n, 1, 63}]

G.f. A(x) satisfies: A(x) = x / (1 - x) + Sum_{k>=2} phi(k) * A(x^k).
a(n) = 1 + Sum_{d|n, d < n} phi(n/d) * a(d).
a(n) = Sum_{d|n} A006874(d).

A326824 a(1) = 1; a(n) = Sum_{d|n, d < n} phi(n/d) * d * a(d).

Original entry on oeis.org

1, 1, 2, 4, 4, 12, 6, 24, 18, 32, 10, 124, 12, 60, 72, 240, 16, 336, 18, 440, 132, 140, 22, 2088, 100, 192, 378, 1044, 28, 2096, 30, 4320, 300, 320, 312, 9636, 36, 396, 408, 10384, 40, 5040, 42, 3500, 3000, 572, 46, 61584, 294, 3920, 672, 5544, 52, 23148, 680

Offset: 1

Ilya Gutkovskiy, Feb 23 2020

nonn

Cf. A000010, A001710, A006579, A006874, A018804, A165552.

a[1] = 1; a[n_] := Sum[If[d < n, EulerPhi[n/d] d a[d], 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 55}]
a[1] = 1; a[n_] := Sum[GCD[n, k] a[GCD[n, k]], {k, 1, n - 1}]; Table[a[n], {n, 1, 55}]

a(1) = 1; a(n) = Sum_{k=1..n-1} gcd(n, k) * a(gcd(n, k)).

A327275 a(1) = 1; a(n) = Sum_{d|n, dA001615.

Original entry on oeis.org

1, 3, 4, 15, 6, 36, 8, 75, 28, 54, 12, 252, 14, 72, 72, 375, 18, 348, 20, 378, 96, 108, 24, 1620, 66, 126, 196, 504, 30, 936, 32, 1875, 144, 162, 144, 3108, 38, 180, 168, 2430, 42, 1248, 44, 756, 696, 216, 48, 9900, 120, 810, 216, 882, 54, 3108, 216, 3240, 240, 270, 60, 8568

Offset: 1

Ilya Gutkovskiy, Sep 15 2019

nonn

Cf. A001615, A006874, A048250, A323363.

a[n_] := If[n == 1, n, Sum[If[d < n, DirichletConvolve[j, MoebiusMu[j]^2, j, n/d] a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 1, 60}]
nmax = 60; A[] = 0; Do[A[x] = x + Sum[DirichletConvolve[j, MoebiusMu[j]^2, j, k] A[x^k], {k, 2, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest

G.f. A(x) satisfies: A(x) = x + Sum_{k>=2} psi(k) * A(x^k).

a(p) = p + 1, where p is prime.

A332778 a(1) = 1; a(n) = Sum_{d|n, d < n} phi(n/d) * a(d)^2.

Original entry on oeis.org

1, 1, 2, 3, 4, 8, 6, 15, 14, 24, 10, 96, 12, 48, 56, 255, 16, 344, 18, 656, 108, 120, 22, 9840, 84, 168, 434, 2448, 28, 4608, 30, 65535, 260, 288, 264, 137376, 36, 360, 360, 432512, 40, 16776, 42, 14720, 7208, 528, 46, 96974880, 258, 9464, 608, 28656, 52, 425864, 600

Offset: 1

Ilya Gutkovskiy, Feb 23 2020

nonn

Cf. A000010, A006874, A007018, A023900, A069097, A082588.

a[1] = 1; a[n_] := Sum[If[d < n, EulerPhi[n/d] a[d]^2, 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 55}]
a[1] = 1; a[n_] := Sum[a[GCD[n, k]]^2, {k, 1, n - 1}]; Table[a[n], {n, 1, 55}]

a(1) = 1; a(n) = Sum_{k=1..n-1} a(gcd(n, k))^2.

A346114 a(n+1) = 1 + Sum_{k=1..n} a(gcd(n,k)).

Original entry on oeis.org

1, 2, 4, 7, 12, 17, 28, 35, 51, 66, 91, 102, 150, 163, 210, 259, 325, 342, 454, 473, 608, 701, 823, 846, 1099, 1168, 1355, 1500, 1786, 1815, 2290, 2321, 2711, 2954, 3328, 3537, 4302, 4339, 4848, 5221, 6075, 6116, 7269, 7312, 8306, 9059, 9949, 9996, 11795, 12006

Offset: 1

Ilya Gutkovskiy, Jul 05 2021

nonn

Cf. A000010, A006874, A038045, A338750.

a[n_] := a[n] = 1 + Sum[a[GCD[n - 1, k]], {k, 1, n - 1}]; Table[a[n], {n, 1, 50}]
nmax = 50; A[] = 0; Do[A[x] = x (1/(1 - x) + Sum[EulerPhi[k] A[x^k], {k, 1, nmax}]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest

G.f. A(x) satisfies: A(x) = x * (1 / (1 - x) + Sum_{k>=1} phi(k) * A(x^k)).

a(1) = 1; a(n+1) = 1 + Sum_{d|n} phi(n/d) * a(d).

cp's OEIS Frontend

A006876 Mu-molecules in Mandelbrot set whose seeds have period n.

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A332791 a(1) = 1; a(n+1) = Sum_{d|n} phi(d) * a(d).

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

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A332792 a(1) = 1; a(n) = Sum_{d|n, d < n} phi(d) * a(d).

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A333613 a(1) = 1; thereafter a(n) = Sum_{k = 1..n} a(k/gcd(n,k)).

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A338750 a(n) = 1 + Sum_{k=1..n-1} a(gcd(n,k)).

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A326824 a(1) = 1; a(n) = Sum_{d|n, d < n} phi(n/d) * d * a(d).

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A327275 a(1) = 1; a(n) = Sum_{d|n, dA001615.

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