A035191 -id:A035191 - cp's OEIS frontend

A326395 Expansion of Sum_{k>=1} x^(2k) (1 + x^k) / (1 - x^(3*k)).

0, 1, 1, 1, 1, 3, 0, 2, 2, 2, 1, 4, 0, 2, 3, 2, 1, 5, 0, 3, 2, 2, 1, 6, 1, 2, 3, 2, 1, 6, 0, 3, 3, 2, 2, 7, 0, 2, 2, 4, 1, 6, 0, 3, 5, 2, 1, 7, 0, 3, 3, 2, 1, 7, 2, 4, 2, 2, 1, 9, 0, 2, 4, 3, 2, 6, 0, 3, 3, 4, 1, 10, 0, 2, 4, 2, 2, 6, 0, 5, 4, 2, 1, 8, 2, 2, 3, 4, 1, 10

Offset: 1

Ilya Gutkovskiy, Sep 11 2019

nonn

Number of divisors of n that are not of the form 3*k + 1.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A001817, A001822, A004611 (positions of 0's), A007494, A035191, A082050, A326394.

Cf. A001620, A256425.

N:= 100: # for a(1) .. a(N)
S:= series(add(x^(2*k)*(1+x^k)/(1-x^(3*k)),k=1..N/2),x,N+1):
seq(coeff(S,x,i),i=1..N); # Robert Israel, Aug 27 2020

nmax = 90; CoefficientList[Series[Sum[x^(2 k) (1 + x^k)/(1 - x^(3 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, 1 &, !MemberQ[{1}, Mod[#, 3]] &], {n, 1, 90}]

a(n) = {numdiv(n) - sumdiv(n, d, d%3==1)} \\ Andrew Howroyd, Sep 11 2019

a(n) = A000005(n) - A001817(n).

Sum_{k=1..n} a(k) ~ 2*n*log(n)/3 + c*n, where c = (5*gamma-2)/3 - gamma(1,3) = (5*A001620-2)/3 - A256425 = -0.382447... . - Amiram Eldar, Jan 14 2024

A327785 Square array read by antidiagonals: A(n,k) = Sum_{d|n} (k/d), (n>=1, k>=0), where (m/n) is the Kronecker symbol.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 0, 0, 3, 1, 1, 1, 1, 1, 2, 1, 1, 0, 2, 1, 0, 4, 1, 1, 1, 0, 1, 0, 0, 2, 1, 1, 2, 1, 1, 2, 0, 2, 4, 1, 1, 1, 2, 1, 1, 2, 0, 1, 3, 1, 1, 2, 0, 3, 2, 0, 2, 0, 1, 4, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 2, 3, 0, 4, 0, 0, 3, 0, 0, 6, 1

Offset: 1

Seiichi Manyama, Sep 25 2019

			Square array begins:
   1, 1, 1, 1, 1, 1, 1, 1, ...
   1, 2, 1, 0, 1, 0, 1, 2, ...
   1, 2, 0, 1, 2, 0, 1, 2, ...
   1, 3, 1, 1, 1, 1, 1, 3, ...
   1, 2, 0, 0, 2, 1, 2, 0, ...
   1, 4, 0, 0, 2, 0, 1, 4, ...
   1, 2, 2, 0, 2, 0, 0, 1, ...
   1, 4, 1, 0, 1, 0, 1, 4, ...

Seiichi Manyama, Antidiagonals n = 1..100, flattened

Columns k=0..31 give A000012, A000005, A035185, A035186, A001227, A035187, A035188, A035189, A035185, A035191, A035192, A035193, A035194, A035195, A035196, A035197, A001227, A035199, A035200, A035201, A035202, A035203, A035204, A035205, A035188, A035207, A035208, A035186, A035210, A035211, A035212, A035213.

Cf. A215200.

A[n_, k_] := Sum[KroneckerSymbol[k, d], {d, Divisors[n]}];
Table[A[n - k, k], {n, 1, 13}, {k, n - 1, 0, -1}] // Flatten (* Jean-François Alcover, Sep 25 2019 *)

A328484 Dirichlet g.f.: zeta(s)^2 / (1 - 3^(-s)).

Original entry on oeis.org

1, 2, 3, 3, 2, 6, 2, 4, 6, 4, 2, 9, 2, 4, 6, 5, 2, 12, 2, 6, 6, 4, 2, 12, 3, 4, 10, 6, 2, 12, 2, 6, 6, 4, 4, 18, 2, 4, 6, 8, 2, 12, 2, 6, 12, 4, 2, 15, 3, 6, 6, 6, 2, 20, 4, 8, 6, 4, 2, 18, 2, 4, 12, 7, 4, 12, 2, 6, 6, 8, 2, 24, 2, 4, 9, 6, 4, 12, 2, 10, 15, 4, 2, 18, 4

Offset: 1

Ilya Gutkovskiy, Oct 16 2019

Inverse Moebius transform of A051064.

Dirichlet convolution of A000005 with characteristic function of powers of 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A051064, A129628.

Cf. A007949, A372713, A373438, A035191.

seq(add(padic[ordp](3*d, 3), d in numtheory[divisors](n)), n=1..100); # Ridouane Oudra, Sep 30 2024

Table[DivisorSum[n, IntegerExponent[3 #, 3] &], {n, 1, 85}]
nmax = 85; CoefficientList[Series[Sum[Sum[x^(i 3^j)/(1 - x^(i 3^j)), {j, 0, Floor[Log[3, nmax]] + 1}], {i, 1, nmax}], {x, 0, nmax}], x] // Rest
f[p_, e_] := If[p == 3, (e + 1)*(e + 2)/2, e + 1]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 02 2020 *)

G.f.: Sum_{i>=1} Sum_{j>=0} x^(i*3^j) / (1 - x^(i*3^j)).
a(n) = Sum_{d|n} A051064(d).
Sum_{k=1..n} a(k) ~ 3*n*(log(n)/2 - log(3)/4 - 1/2 + gamma), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 17 2019
Multiplicative with a(p^e) = (e+1)*(e+2)/2 if p=3, and e+1 otherwise. - Amiram Eldar, Dec 02 2020
From Ridouane Oudra, Sep 30 2024: (Start)
a(n) = Sum_{i=0..A007949(n)} tau(n/3^i).
a(n) = Sum_{d|3*n} A007949(d).
a(n) = (1/2)*A051064(n)*A372713(n).
a(n) = (1/2)*(A051064(n) + 1)*A000005(n).
a(n) = A373438(n)*A035191(n). (End)

cp's OEIS Frontend

A326395 Expansion of Sum_{k>=1} x^(2k) (1 + x^k) / (1 - x^(3*k)).

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A327785 Square array read by antidiagonals: A(n,k) = Sum_{d|n} (k/d), (n>=1, k>=0), where (m/n) is the Kronecker symbol.

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A328484 Dirichlet g.f.: zeta(s)^2 / (1 - 3^(-s)).

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cp's OEIS Frontend

A326395 Expansion of Sum_{k>=1} x^(2*k) * (1 + x^k) / (1 - x^(3*k)).

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A327785 Square array read by antidiagonals: A(n,k) = Sum_{d|n} (k/d), (n>=1, k>=0), where (m/n) is the Kronecker symbol.

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A328484 Dirichlet g.f.: zeta(s)^2 / (1 - 3^(-s)).

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Author

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Programs

Maple

Mathematica

Formula

A326395 Expansion of Sum_{k>=1} x^(2k) (1 + x^k) / (1 - x^(3*k)).