A195459 -id:A195459 - cp's OEIS frontend

A327625 Expansion of Sum_{k>=0} x^(3^k) / (1 - x^(3^k))^2.

1, 2, 4, 4, 5, 8, 7, 8, 13, 10, 11, 16, 13, 14, 20, 16, 17, 26, 19, 20, 28, 22, 23, 32, 25, 26, 40, 28, 29, 40, 31, 32, 44, 34, 35, 52, 37, 38, 52, 40, 41, 56, 43, 44, 65, 46, 47, 64, 49, 50, 68, 52, 53, 80, 55, 56, 76, 58, 59, 80, 61, 62, 91, 64, 65, 88, 67, 68, 92, 70

Offset: 1

Ilya Gutkovskiy, Sep 19 2019

Sum of divisors d of n such that n/d is power of 3.

Inverse Moebius transform of A195459.

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

Cf. A000010, A000244, A001651 (fixed points), A051064, A129527, A195459.

[(1/2)*&+[EulerPhi(3*d) :d in Divisors(n)]:n in [1..70]]; // Marius A. Burtea, Sep 19 2019

nmax = 70; CoefficientList[Series[Sum[x^(3^k)/(1 - x^(3^k))^2, {k, 0, Floor[Log[3, nmax]] + 1}], {x, 0, nmax}], x] // Rest
a[n_] := DivisorSum[n, # &, IntegerQ[Log[3, n/#]] &]; Table[a[n], {n, 1, 70}]
a[n_] := 1/2 Sum[EulerPhi[3 d], {d, Divisors[n]}]; Table[a[n], {n, 1, 70}]

A327625(n) = (n+sumdiv(n,d,my(b=0); if(isprimepower(n/d,&b)&&(3==b),d,0))); \\ Antti Karttunen, Sep 19 2019

G.f. A(x) satisfies: A(x) = A(x^3) + x/(1 - x)^2.
G.f.: Sum_{k>=1} phi(3*k) * x^k / (2 * (1 - x^k)), where phi = A000010.
a(n) = (1/2) * Sum_{d|n} phi(3*d).
From Amiram Eldar, Nov 17 2022: (Start)
Multiplicative with a(3^e) = (3^(e+1)-1)/2, and a(p^e) = p^e for p != 3.
Sum_{k=1..n} a(k) ~ (9/16) * n^2. (End)
Dirichlet g.f.: zeta(s-1)*(1+1/(3^s-1)). - Amiram Eldar, Dec 17 2022

A372673 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = phi(k*n) / phi(k).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 2, 3, 4, 4, 1, 1, 2, 2, 4, 2, 1, 2, 2, 4, 4, 4, 6, 1, 1, 3, 2, 4, 3, 6, 4, 1, 2, 2, 4, 5, 4, 6, 8, 6, 1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 1, 2, 3, 4, 4, 6, 6, 8, 9, 8, 10, 1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 1, 2, 2, 4, 4, 4, 7, 8, 6, 8, 10, 8, 12, 1, 1, 3, 2, 5, 3, 6, 4, 9, 5, 10, 6, 12, 6

Offset: 1

Seiichi Manyama, May 10 2024

			Square array T(n,k) begins:
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, ...
  2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, ...
  2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, ...
  4, 4, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 4, ...
  2, 4, 3, 4, 2, 6, 2, 4, 3, 4, 2, 6, 2, 4, 3, 4, 2, 6, ...
  6, 6, 6, 6, 6, 6, 7, 6, 6, 6, 6, 6, 6, 7, 6, 6, 6, 6, ...
  4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, ...
  6, 6, 9, 6, 6, 9, 6, 6, 9, 6, 6, 9, 6, 6, 9, 6, 6, 9, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Columns k=1..18 give: A000010, A062570, A195459, A062570, A359101, A372671, A359102, A062570, A195459, A372672, A372676, A372671, A372677, A372678, A372679, A062570, A372681, A372671.
Main diagonal gives A000027.
Cf. A005117, A372619.

PARI
```
T(n, k) = eulerphi(k*n)/eulerphi(k);
```

A359101 a(n) = phi(5 * n)/4.

Original entry on oeis.org

1, 1, 2, 2, 5, 2, 6, 4, 6, 5, 10, 4, 12, 6, 10, 8, 16, 6, 18, 10, 12, 10, 22, 8, 25, 12, 18, 12, 28, 10, 30, 16, 20, 16, 30, 12, 36, 18, 24, 20, 40, 12, 42, 20, 30, 22, 46, 16, 42, 25, 32, 24, 52, 18, 50, 24, 36, 28, 58, 20, 60, 30, 36, 32, 60, 20, 66, 32, 44, 30, 70, 24, 72, 36, 50, 36, 60, 24, 78

Offset: 1

Seiichi Manyama, Dec 16 2022

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Moebius Transform.
Eric Weisstein's World of Mathematics, Totient Function.

Cf. A195459, A359102.

Cf. A008653, A359100.

Array[EulerPhi[5 #]/4 &, 79] (* Michael De Vlieger, Dec 16 2022 *)

PARI
```
a(n) = eulerphi(5*n)/4;
```

my(N=80, x='x+O('x^N)); Vec(-sum(k=1, N, moebius(5*k)*x^k/(1-x^k)^2))

G.f.: -Sum_{k>=1} mu(5 * k) * x^k / (1 - x^k)^2, where mu() is the Moebius function (A008683).
From Amiram Eldar, Dec 17 2022: (Start)
Multiplicative with a(5^e) = 5^e, and a(p^e) = (p-1)*p^(e-1) if p != 5.
Dirichlet g.f.: zeta(s-1)/(zeta(s)*(1-1/5^s)).
Sum_{k=1..n} a(k) ~ (25/(8*Pi^2)) * n^2. (End)

A359102 a(n) = phi(7 * n)/6.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 7, 4, 6, 4, 10, 4, 12, 7, 8, 8, 16, 6, 18, 8, 14, 10, 22, 8, 20, 12, 18, 14, 28, 8, 30, 16, 20, 16, 28, 12, 36, 18, 24, 16, 40, 14, 42, 20, 24, 22, 46, 16, 49, 20, 32, 24, 52, 18, 40, 28, 36, 28, 58, 16, 60, 30, 42, 32, 48, 20, 66, 32, 44, 28, 70, 24, 72, 36, 40, 36, 70, 24, 78

Offset: 1

Seiichi Manyama, Dec 16 2022

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Moebius Transform.
Eric Weisstein's World of Mathematics, Totient Function.

Cf. A195459, A359101.

Cf. A008653, A359099.

Array[EulerPhi[7 #]/6 &, 79] (* Michael De Vlieger, Dec 16 2022 *)

PARI
```
a(n) = eulerphi(7*n)/6;
```

my(N=80, x='x+O('x^N)); Vec(-sum(k=1, N, moebius(7*k)*x^k/(1-x^k)^2))

G.f.: -Sum_{k>=1} mu(7 * k) * x^k / (1 - x^k)^2, where mu() is the Moebius function (A008683).
From Amiram Eldar, Dec 17 2022: (Start)
Multiplicative with a(7^e) = 7^e, and a(p^e) = (p-1)*p^(e-1) if p != 7.
Dirichlet g.f.: zeta(s-1)/(zeta(s)*(1-1/7^s)).
Sum_{k=1..n} a(k) ~ (49/(16*Pi^2)) * n^2. (End)

A194881 A number of sum-free sets related to fractional parts of multiples of a rational number in the range 1/3 to 2/3.

Original entry on oeis.org

2, 3, 6, 8, 12, 15, 21, 25, 34, 38, 48, 54, 66, 72, 84, 92, 108, 117, 135, 143, 161, 171, 193, 205, 225, 237, 264, 276, 304, 316, 346, 362, 392, 408, 432, 450, 486, 504, 540, 556, 596, 614, 656, 676, 712, 734, 780, 804, 846

Offset: 1

R. J. Mathar, Sep 04 2011

nonn

Deserves a better title which mentions n in the sense that this is a sum-free set from a difference set with {1,....,n}.

Peter J. Cameron and Paul Erdős, Notes on sum-free and related sets, Combinat. Probabl. Comput. 8 (1&2) (1999), 95-107, Theorem 4.

Cf. A000010, A195459, A372621.

A194881 := proc(n) 1+add(numtheory[phi](3*q),q=1..n)/2 ; end proc:
seq(A194881(n),n=1..80) ;

Accumulate[Table[EulerPhi[3*n], {n, 1, 60}]]/2 + 1 (* Amiram Eldar, May 08 2024 *)

a(n) = 1 + Sum_{j=1..n} A000010(3*j)/2.

a(n) ~ (27/(8*Pi^2)) * n^2. - Amiram Eldar, May 08 2024

A227128 The twisted Euler phi-function for the non-principal Dirichlet character mod 3.

Original entry on oeis.org

1, 3, 3, 6, 6, 9, 6, 12, 9, 18, 12, 18, 12, 18, 18, 24, 18, 27, 18, 36, 18, 36, 24, 36, 30, 36, 27, 36, 30, 54, 30, 48, 36, 54, 36, 54, 36, 54, 36, 72, 42, 54, 42, 72, 54, 72, 48, 72, 42, 90, 54, 72, 54, 81, 72, 72, 54, 90, 60, 108, 60, 90, 54, 96

Offset: 1

R. J. Mathar, Jul 02 2013

The non-principal Dirichlet character mod 3 is chi(n) = A049347(n-1). The twisted Euler phi-function is defined as a(n) = phi(n,chi) = n*Product_{p|n} (1-chi(p)/p), where the product is over all primes p that divide n.
The sequence appears to be the Dirichlet convolution of the sequence A055615(n) and a sequence of signed 1's with the same characteristic function as A156277.
Sequences phi(n,chi) are defined as well for chi=A101455, chi=A080891, chi=A134667 and so on.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Olivier Bordellès and Benoit Cloitre, An alternating sum involving the reciprocal of certain multiplicative functions, J. Int. Seq. 16 (2013), Article 13.6.3.
Jerzy Kaczorowski and Kazimierz Wiertelak, On the sum of the twisted Euler function, Int. J. Numb. Theory 8 (7) (2012), 1741-1761.

Cf. A049347, A080891, A086724, A101455, A134667, A195459 (for the principal character mod 3), A227128.

chi := proc(n)
    op(1+(n mod 3),[0,1,-1]) ;
end proc:
A227128 := proc(n)
    local a,p ;
    a := n ;
    for p in numtheory[factorset](n) do
        a := a*(1-chi(p)/p) ;
    end do:
    a ;
end proc:

f[p_, e_] := If[Mod[p, 3] == 2, p + 1, p - 1]*p^(e - 1); f[3, e_] := 3^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 13 2022 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] == 3, 3^f[i,2], f[i, 1]^(f[i,2] - 1) * (f[i,1] + (-1)^(f[i,1]%3))))}; \\ Amiram Eldar, Oct 13 2022

Multiplicative with a(3^e) = 3^e, a(p^e) = p^(e-1)*(p-1) if p == 1 (mod 3) and a(p^e) = p^(e-1)*(p+1) if p == 2 (mod 3). - R. J. Mathar, Jul 10 2013
From Amiram Eldar, Oct 13 2022: (Start)
a(n) = A227128(n)/2 if n divisible by 3, and a(n) = A227128(n) otherwise.
Sum_{k=1..n} a(k) ~ c * n^2, where c = 1/(2 * A086724) = 0.639957... . (End)

A364210 a(n) = (1/(2n)) Sum_{d|n} 3^(n/d-1) * phi(3*d).

Original entry on oeis.org

1, 2, 4, 8, 17, 44, 105, 278, 733, 1978, 5369, 14792, 40881, 113934, 318884, 896948, 2532161, 7174862, 20390553, 58114072, 166037460, 475473286, 1364393897, 3922640132, 11297181473, 32588043882, 94143179560, 272342824320, 788854912241, 2287679406940, 6641649422409

Offset: 1

Seiichi Manyama, Jul 13 2023

nonn

Cf. A000010, A034754, A195459, A327625.

Cf. A000013, A364211, A364212.

a[n_] := DivisorSum[n, 3^(n/#-1)*EulerPhi[3*#]/(2*n) &]; Array[a, 30] (* Amiram Eldar, Jul 14 2023 *)

a(n) = sumdiv(n, d, 3^(n/d-1)*eulerphi(3*d))/(2*n);

G.f.: (-1/2) * Sum_{k>0} phi(3*k) * log(1-3*x^k)/(3*k).

A372621 a(n) = (1/2) * Sum_{k=1..n} phi(3*k).

Original entry on oeis.org

1, 2, 5, 7, 11, 14, 20, 24, 33, 37, 47, 53, 65, 71, 83, 91, 107, 116, 134, 142, 160, 170, 192, 204, 224, 236, 263, 275, 303, 315, 345, 361, 391, 407, 431, 449, 485, 503, 539, 555, 595, 613, 655, 675, 711, 733, 779, 803, 845, 865, 913, 937, 989, 1016, 1056, 1080, 1134

Offset: 1

Seiichi Manyama, May 07 2024

nonn

Eric Weisstein's World of Mathematics, Totient Function.

Equals A194881 - 1.
Partial sums of A195459.
Column k=3 of A372619.
Cf. A000010, A194881.

Accumulate[Table[EulerPhi[3*n], {n, 1, 60}]]/2 (* Amiram Eldar, May 08 2024 *)

PARI
```
a(n) = sum(k=1, n, eulerphi(3*k))/2;
```

a(n) ~ (27/(8*Pi^2)) * n^2. - Amiram Eldar, May 08 2024

cp's OEIS Frontend

A327625 Expansion of Sum_{k>=0} x^(3^k) / (1 - x^(3^k))^2.

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Magma

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A372673 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = phi(k*n) / phi(k).

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A359101 a(n) = phi(5 * n)/4.

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Mathematica

PARI

PARI

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A359102 a(n) = phi(7 * n)/6.

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PARI

PARI

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A194881 A number of sum-free sets related to fractional parts of multiples of a rational number in the range 1/3 to 2/3.

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Programs

Maple

Mathematica

Formula

A227128 The twisted Euler phi-function for the non-principal Dirichlet character mod 3.

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Programs

Maple

Mathematica

PARI

Formula

A364210 a(n) = (1/(2*n)) * Sum_{d|n} 3^(n/d-1) * phi(3*d).

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Mathematica

PARI

Formula

A372621 a(n) = (1/2) * Sum_{k=1..n} phi(3*k).

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A364210 a(n) = (1/(2n)) Sum_{d|n} 3^(n/d-1) * phi(3*d).