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.

A130055 A129691 * A127093.

Original entry on oeis.org

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

Views

Author

Gary W. Adamson, May 04 2007

Keywords

Comments

Row sums = d(n), A000005: (1, 2, 2, 3, 2, 4, 2, 4, 3, 4, ...). Left column = A130054: (1, 0, -1, -1, -3, 0, -5, -2, -3, 0, ...).

Examples

			First few rows of the triangle:
   1;
   0,  2;
  -1,  0,  3;
  -1,  0,  0,  4;
  -3,  0,  0,  0,  5;
   0, -2,  0,  0,  0,  6;
  -5,  0,  0,  0,  0,  0,  7;
  ...

Crossrefs

Cf. A129691, A127093, A000005, A130054.

Formula

A129691 * A127093 as infinite lower triangular matrices.

A128980 A054525 * A129691(unsigned).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 3, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 5, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 3, 0, 0, 0, 0, 0, 0, 0, 1
Offset: 1

Views

Author

Gary W. Adamson, Apr 29 2007

Keywords

Comments

Row sums = A070777: (1, 1, 2, 1, 4, 2, 6, 1, 2, 4, ...). A129691 = the unsigned inverse of A054523.

Examples

			First few rows of the triangle:
  1;
  0, 1;
  1, 0, 1;
  0, 0, 0, 1;
  3, 0, 0, 0, 1;
  0, 1, 0, 0, 0, 1;
  5, 0, 0, 0, 0, 0, 1;
  0, 0, 0, 0, 0, 0, 0, 1;
  0, 0, 1, 0, 0, 0, 0, 0, 1;
  ...

Crossrefs

Cf. A054525, A129691, A070777, A054523.

Formula

Moebius transform of A129691

A130054 Inverse Moebius transform of A023900.

Original entry on oeis.org

1, 0, -1, -1, -3, 0, -5, -2, -3, 0, -9, 1, -11, 0, 3, -3, -15, 0, -17, 3, 5, 0, -21, 2, -7, 0, -5, 5, -27, 0, -29, -4, 9, 0, 15, 3, -35, 0, 11, 6, -39, 0, -41, 9, 9, 0, -45, 3, -11, 0, 15, 11, -51, 0, 27, 10, 17, 0, -57, -3, -59, 0, 15, -5, 33, 0, -65, 15, 21
Offset: 1

Views

Author

Gary W. Adamson, May 04 2007

Keywords

Comments

Multiplicative because A023900 is. - Andrew Howroyd, Aug 03 2018

Links

Crossrefs

Row sums of A129691.
Cf. A126988, A000005, A023900, A062570, A001511, A018804, A000203, A007431.

Programs

  • Magma
    [&+[d*MoebiusMu(d)*NumberOfDivisors(n div d):d in Divisors(n)]:n in [1..70]]; // Marius A. Burtea, Nov 17 2019
  • Maple
    with(numtheory): seq(add(d*mobius(d)*tau(n/d), d in divisors(n)), n=1..60); # Ridouane Oudra, Nov 17 2019
  • Mathematica
    b[n_] := Sum[d MoebiusMu[d], {d, Divisors[n]}];
a[n_] := Sum[b[n/d], {d, Divisors[n]}];
a /@ Range[1, 100] (* Jean-François Alcover, Sep 20 2019, from PARI *)
f[p_, e_] := 1-(p-1)*e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 23 2020 *)
  • PARI
    \\ here b(n) is A023900
b(n)={sumdivmult(n, d, d*moebius(d))}
a(n)={sumdiv(n, d, b(n/d))} \\ Andrew Howroyd, Aug 03 2018

Formula

A126988 * A130054 = d(n), A000005: (1, 2, 2, 3, 2, 4, 2, 4, 3, 4, ...).
a(n) = Sum_{d|n} A023900(n/d). - Andrew Howroyd, Aug 03 2018
a(n) = Sum_{d|n} d*mu(d)*tau(n/d). - Ridouane Oudra, Nov 17 2019
From Werner Schulte, Sep 06 2020: (Start)
Multiplicative with a(p^e) = 1 - (p-1) * e for prime p and e >= 0.
Dirichlet g.f.: (zeta(s))^2 / zeta(s-1).
Dirichlet convolution with A062570 equals A001511.
Dirichlet convolution with A018804 equals A000203.
Dirichlet inverse of A007431. (End)
a(n) = 1 - Sum_{k=1..n-1} a(gcd(n,k)). - Ilya Gutkovskiy, Nov 06 2020

Extensions

Name changed and terms a(11) and beyond from Andrew Howroyd, Aug 03 2018

A130029 a(n) = Sum_{d|n} phi(n/d) * prime(d).

Original entry on oeis.org

2, 5, 9, 14, 19, 28, 29, 40, 45, 60, 51, 88, 65, 90, 105, 114, 91, 150, 103, 178, 161, 160, 127, 252, 181, 202, 215, 268, 165, 352, 187, 306, 289, 278, 331, 462, 229, 320, 357, 506, 259, 542, 275, 474, 537, 392, 303, 706, 413, 586, 495, 590, 345, 720, 571, 764, 565, 520
Offset: 1

Views

Author

Gary W. Adamson, May 02 2007

Keywords

Comments

Old name: A054523 * A000040.

Examples

			a(4) = 14 = dot product of row 4 of A054523, (2, 1, 0, 1) and primes (2, 3, 5, 7) = (4 + 3 + 0 + 7) = 14.

Crossrefs

Cf. A000010, A000040, A054523, A129691, A130030.

Programs

Formula

A054523 as an infinite lower triangular matrix * A000040 (the primes) as a vector.
a(n) = Sum_{k=1..n} prime(gcd(n,k)). - Ilya Gutkovskiy, Mar 22 2020

Extensions

New name and more terms from Ilya Gutkovskiy, Mar 22 2020

A130030 a(n) = Sum_{d|n} (-1)^omega(n/d) * phi(rad(n/d)) * prime(d).

Original entry on oeis.org

2, 1, 1, 2, 3, 6, 5, 7, 9, 14, 11, 15, 17, 20, 21, 22, 27, 20, 31, 23, 33, 38, 39, 20, 45, 48, 43, 35, 53, 6, 67, 47, 65, 64, 63, 25, 85, 78, 73, 34, 99, 20, 107, 63, 45, 94, 119, 35, 113, 56, 99, 73, 137, 54, 103, 54, 117, 134, 161, -1, 163, 136, 73, 96, 113, 24, 199, 107, 159, 12, 213
Offset: 1

Views

Author

Gary W. Adamson, May 02 2007

Keywords

Comments

Old name: A129691 * A000040.
A130029 = A054523 * A000040.

Examples

			a(4) = 2 = dot product of row 4 of A129691: (-1, -1, 0, 1) and the first four primes: (2, 3, 5, 7) = (-2, -3, 0, 7) = 2.

Crossrefs

Cf. A000010, A000040, A001221, A007444, A007947, A023900, A054523, A076479, A129691, A130029, A333177.

Formula

A129691 as an infinite lower triangular matrix * A000040, the primes.
From Ilya Gutkovskiy, Mar 22 2020: (Start)
a(n) = Sum_{d|n} A023900(n/d) * prime(d).
Sum_{k=1..n} a(gcd(n,k)) = prime(n). (End)

Extensions

New name and more terms from Ilya Gutkovskiy, Mar 22 2020

A366444 Triangle read by rows: T(n,k) = phi(n/k)*A023900(k) if k divides n, T(n,k) = 0 otherwise (n >= 1, 1 <= k <= n).

Original entry on oeis.org

1, 1, -1, 2, 0, -2, 2, -1, 0, -1, 4, 0, 0, 0, -4, 2, -2, -2, 0, 0, 2, 6, 0, 0, 0, 0, 0, -6, 4, -2, 0, -1, 0, 0, 0, -1, 6, 0, -4, 0, 0, 0, 0, 0, -2, 4, -4, 0, 0, -4, 0, 0, 0, 0, 4, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, -10, 4, -2, -4, -2, 0, 2, 0, 0, 0, 0, 0, 2
Offset: 1

Views

Author

Mats Granvik, Oct 12 2023

Keywords

Comments

Sum_{k=1..n} T(n,k) = A063524(n).

Examples

			{
{1}, = 1
{1, -1}, = 0
{2, 0, -2}, = 0
{2, -1, 0, -1}, = 0
{4, 0, 0, 0, -4}, = 0
{2, -2, -2, 0, 0, 2}, = 0
{6, 0, 0, 0, 0, 0, -6}, = 0
{4, -2, 0, -1, 0, 0, 0, -1}, = 0
{6, 0, -4, 0, 0, 0, 0, 0, -2}, = 0
{4, -4, 0, 0, -4, 0, 0, 0, 0, 4}, = 0
{10, 0, 0, 0, 0, 0, 0, 0, 0, 0, -10}, = 0
{4, -2, -4, -2, 0, 2, 0, 0, 0, 0, 0, 2} = 0
}

Crossrefs

Cf. A000010, A023900, A366445, A054524, A054525, A063524, A054522, A054523, A129691, A127649.

Programs

  • Mathematica
    nn = 12; g[n_] := DivisorSum[n, MoebiusMu[#] # &]; Flatten[Table[Table[If[Mod[n, k] == 0, EulerPhi[n/k]*g[k], 0], {k, 1, n}], {n, 1, nn}]]

Formula

T(n,k) = phi(n/k)*A023900(k) if k divides n, T(n,k) = 0 otherwise (n >= 1, 1 <= k <= n).

A366445 Triangle read by rows: T(n,k) = A023900(n/k)*phi(k) if k divides n, T(n,k) = 0 otherwise (n >= 1, 1 <= k <= n).

Original entry on oeis.org

1, -1, 1, -2, 0, 2, -1, -1, 0, 2, -4, 0, 0, 0, 4, 2, -2, -2, 0, 0, 2, -6, 0, 0, 0, 0, 0, 6, -1, -1, 0, -2, 0, 0, 0, 4, -2, 0, -4, 0, 0, 0, 0, 0, 6, 4, -4, 0, 0, -4, 0, 0, 0, 0, 4, -10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 2, 2, -2, -4, 0, -2, 0, 0, 0, 0, 0, 4
Offset: 1

Views

Author

Mats Granvik, Oct 12 2023

Keywords

Comments

Sum_{k=1..n} T(n,k) = A063524(n).

Examples

			{
{1}, = 1
{-1, 1}, = 0
{-2, 0, 2}, = 0
{-1, -1, 0, 2}, = 0
{-4, 0, 0, 0, 4}, = 0
{2, -2, -2, 0, 0, 2}, = 0
{-6, 0, 0, 0, 0, 0, 6}, = 0
{-1, -1, 0, -2, 0, 0, 0, 4}, = 0
{-2, 0, -4, 0, 0, 0, 0, 0, 6}, = 0
{4, -4, 0, 0, -4, 0, 0, 0, 0, 4}, = 0
{-10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10}, = 0
{2, 2, -2, -4, 0, -2, 0, 0, 0, 0, 0, 4} = 0
}

Crossrefs

Cf. A000010, A023900, A366444, A054524, A054525, A063524, A054522, A054523, A129691, A127649.

Programs

  • Mathematica
    nn = 12; g[n_] := DivisorSum[n, MoebiusMu[#] # &]; Flatten[Table[Table[If[Mod[n, k] == 0, g[n/k]*EulerPhi[k], 0], {k, 1, n}], {n, 1, nn}]]

Formula

T(n,k) = A023900(n/k)*phi(k) if k divides n, T(n,k) = 0 otherwise (n >= 1, 1 <= k <= n).
Showing 1-7 of 7 results.

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