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 1-7 of 7 results.

A341831 Dirichlet g.f.: 1 / zeta(s)^5.

Original entry on oeis.org

1, -5, -5, 10, -5, 25, -5, -10, 10, 25, -5, -50, -5, 25, 25, 5, -5, -50, -5, -50, 25, 25, -5, 50, 10, 25, -10, -50, -5, -125, -5, -1, 25, 25, 25, 100, -5, 25, 25, 50, -5, -125, -5, -50, -50, 25, -5, -25, 10, -50, 25, -50, -5, 50, 25, 50, 25, 25, -5, 250, -5, 25, -50, 0, 25
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 21 2021

Keywords

Comments

Dirichlet inverse of A061200.

Links

Crossrefs

Row 5 of A346148.
Cf. A007427, A007428, A008683, A061200, A247343, A341832, A341833, A341834, A341835, A341836.

Programs

  • Mathematica
    a[1] = 1; a[n_] := Times @@ ((-1)^#[[2]] Binomial[5, #[[2]]] &/@ FactorInteger[n]); Table[a[n], {n, 65}]
  • PARI
    for(n=1, 100, print1(direuler(p=2, n, (1 - X)^5)[n], ", ")) \\ Vaclav Kotesovec, Feb 22 2021

Formula

Multiplicative with a(p^e) = (-1)^e * binomial(5, e).
a(1) = 1; a(n) = -Sum_{d|n, d < n} tau_5(n/d) * a(d).

A341832 Dirichlet g.f.: 1 / zeta(s)^6.

Original entry on oeis.org

1, -6, -6, 15, -6, 36, -6, -20, 15, 36, -6, -90, -6, 36, 36, 15, -6, -90, -6, -90, 36, 36, -6, 120, 15, 36, -20, -90, -6, -216, -6, -6, 36, 36, 36, 225, -6, 36, 36, 120, -6, -216, -6, -90, -90, 36, -6, -90, 15, -90, 36, -90, -6, 120, 36, 120, 36, 36, -6, 540, -6, 36, -90, 1, 36
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 21 2021

Keywords

Comments

Dirichlet inverse of A034695.

Links

Crossrefs

Row 6 of A346148.
Cf. A007427, A007428, A008683, A034695, A247343, A341831, A341833, A341834, A341835, A341836.

Programs

  • Mathematica
    a[1] = 1; a[n_] := Times @@ ((-1)^#[[2]] Binomial[6, #[[2]]] &/@ FactorInteger[n]); Table[a[n], {n, 65}]
  • PARI
    for(n=1, 100, print1(direuler(p=2, n, (1 - X)^6)[n], ", ")) \\ Vaclav Kotesovec, Feb 22 2021

Formula

Multiplicative with a(p^e) = (-1)^e * binomial(6, e).
a(1) = 1; a(n) = -Sum_{d|n, d < n} tau_6(n/d) * a(d).

A341833 Dirichlet g.f.: 1 / zeta(s)^7.

Original entry on oeis.org

1, -7, -7, 21, -7, 49, -7, -35, 21, 49, -7, -147, -7, 49, 49, 35, -7, -147, -7, -147, 49, 49, -7, 245, 21, 49, -35, -147, -7, -343, -7, -21, 49, 49, 49, 441, -7, 49, 49, 245, -7, -343, -7, -147, -147, 49, -7, -245, 21, -147, 49, -147, -7, 245, 49, 245, 49, 49, -7, 1029, -7, 49
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 21 2021

Keywords

Comments

Dirichlet inverse of A111217.

Links

Crossrefs

Row 7 of A346148.
Cf. A007427, A007428, A008683, A111217, A247343, A341831, A341832, A341834, A341835, A341836.

Programs

  • Mathematica
    a[1] = 1; a[n_] := Times @@ ((-1)^#[[2]] Binomial[7, #[[2]]] &/@ FactorInteger[n]); Table[a[n], {n, 62}]
  • PARI
    for(n=1, 100, print1(direuler(p=2, n, (1 - X)^7)[n], ", ")) \\ Vaclav Kotesovec, Feb 22 2021

Formula

Multiplicative with a(p^e) = (-1)^e * binomial(7, e).
a(1) = 1; a(n) = -Sum_{d|n, d < n} tau_7(n/d) * a(d).

A341834 Dirichlet g.f.: 1 / zeta(s)^8.

Original entry on oeis.org

1, -8, -8, 28, -8, 64, -8, -56, 28, 64, -8, -224, -8, 64, 64, 70, -8, -224, -8, -224, 64, 64, -8, 448, 28, 64, -56, -224, -8, -512, -8, -56, 64, 64, 64, 784, -8, 64, 64, 448, -8, -512, -8, -224, -224, 64, -8, -560, 28, -224, 64, -224, -8, 448, 64, 448, 64, 64, -8, 1792
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 21 2021

Keywords

Comments

Dirichlet inverse of A111218.

Links

Crossrefs

Row 8 of A346148.
Cf. A007427, A007428, A008683, A111218, A247343, A341831, A341832, A341833, A341835, A341836.

Programs

  • Mathematica
    a[1] = 1; a[n_] := Times @@ ((-1)^#[[2]] Binomial[8, #[[2]]] &/@ FactorInteger[n]); Table[a[n], {n, 60}]
  • PARI
    for(n=1, 100, print1(direuler(p=2, n, (1 - X)^8)[n], ", ")) \\ Vaclav Kotesovec, Feb 22 2021

Formula

Multiplicative with a(p^e) = (-1)^e * binomial(8, e).
a(1) = 1; a(n) = -Sum_{d|n, d < n} tau_8(n/d) * a(d).

A341835 Dirichlet g.f.: 1 / zeta(s)^9.

Original entry on oeis.org

1, -9, -9, 36, -9, 81, -9, -84, 36, 81, -9, -324, -9, 81, 81, 126, -9, -324, -9, -324, 81, 81, -9, 756, 36, 81, -84, -324, -9, -729, -9, -126, 81, 81, 81, 1296, -9, 81, 81, 756, -9, -729, -9, -324, -324, 81, -9, -1134, 36, -324, 81, -324, -9, 756, 81, 756, 81, 81
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 21 2021

Keywords

Comments

Dirichlet inverse of A111219.

Links

Crossrefs

Row 9 of A346148.
Cf. A007427, A007428, A008683, A111219, A247343, A341831, A341832, A341833, A341834, A341836.

Programs

  • Mathematica
    a[1] = 1; a[n_] := Times @@ ((-1)^#[[2]] Binomial[9, #[[2]]] &/@ FactorInteger[n]); Table[a[n], {n, 58}]
  • PARI
    for(n=1, 100, print1(direuler(p=2, n, (1 - X)^9)[n], ", ")) \\ Vaclav Kotesovec, Feb 22 2021

Formula

Multiplicative with a(p^e) = (-1)^e * binomial(9, e).
a(1) = 1; a(n) = -Sum_{d|n, d < n} tau_9(n/d) * a(d).

A341836 Dirichlet g.f.: 1 / zeta(s)^10.

Original entry on oeis.org

1, -10, -10, 45, -10, 100, -10, -120, 45, 100, -10, -450, -10, 100, 100, 210, -10, -450, -10, -450, 100, 100, -10, 1200, 45, 100, -120, -450, -10, -1000, -10, -252, 100, 100, 100, 2025, -10, 100, 100, 1200, -10, -1000, -10, -450, -450, 100, -10, -2100, 45, -450, 100, -450
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 21 2021

Keywords

Comments

Dirichlet inverse of A111220.

Links

Crossrefs

Row 10 of A346148.
Cf. A007427, A007428, A008683, A111220, A247343, A341831, A341832, A341833, A341834, A341835.

Programs

  • Mathematica
    a[1] = 1; a[n_] := Times @@ ((-1)^#[[2]] Binomial[10, #[[2]]] &/@ FactorInteger[n]); Table[a[n], {n, 52}]
  • PARI
    for(n=1, 100, print1(direuler(p=2, n, (1 - X)^10)[n], ", ")) \\ Vaclav Kotesovec, Feb 22 2021

Formula

Multiplicative with a(p^e) = (-1)^e * binomial(10, e).
a(1) = 1; a(n) = -Sum_{d|n, d < n} tau_10(n/d) * a(d).

A341837 If n = Product (p_j^k_j) then a(n) = Product ((-1)^k_j * binomial(n, k_j)).

Original entry on oeis.org

1, -2, -3, 6, -5, 36, -7, -56, 36, 100, -11, -792, -13, 196, 225, 1820, -17, -2754, -19, -3800, 441, 484, -23, 48576, 300, 676, -2925, -10584, -29, -27000, -31, -201376, 1089, 1156, 1225, 396900, -37, 1444, 1521, 395200, -41, -74088, -43, -41624, -44550
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 21 2021

Keywords

Links

Crossrefs

Cf. A007427, A007428, A008683, A163767, A247343, A341831, A341832, A341833, A341834, A341835, A341836.
Cf. A346148.

Programs

  • Mathematica
    a[1] = 1; a[n_] := Times @@ ((-1)^#[[2]] Binomial[n, #[[2]]] &/@ FactorInteger[n]); Table[a[n], {n, 45}]
  • PARI
    a(n) = my(f=factor(n)[,2]); prod(k=1, #f, (-1)^f[k]*binomial(n, f[k])); \\ Michel Marcus, Feb 21 2021

Formula

a(n) = A346148(n, n). - Sebastian Karlsson, Aug 22 2021
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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