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 1-3 of 3 results.

A338164 Dirichlet g.f.: (zeta(s-2) / zeta(s))^2.

Original entry on oeis.org

1, 6, 16, 33, 48, 96, 96, 168, 208, 288, 240, 528, 336, 576, 768, 816, 576, 1248, 720, 1584, 1536, 1440, 1056, 2688, 1776, 2016, 2448, 3168, 1680, 4608, 1920, 3840, 3840, 3456, 4608, 6864, 2736, 4320, 5376, 8064, 3360, 9216, 3696, 7920, 9984, 6336, 4416, 13056, 7008, 10656
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 14 2020

Keywords

Comments

Dirichlet convolution of Jordan function J_2 (A007434) with itself.

Links

Crossrefs

Cf. A000005, A007427, A007434, A029935, A034714, A306379, A321322, A338165.

Programs

  • Mathematica
    Jordan2[n_] := Sum[d^2 MoebiusMu[n/d], {d, Divisors[n]}]; a[n_] := Sum[Jordan2[d] Jordan2[n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 50}]
a[1] = 1; f[p_, e_] := p^(2 e - 4) (p^4 + e (p^2 - 1)^2 - 1); a[n_] := Times @@ f @@@ FactorInteger[n]; Table[a[n], {n, 1, 50}]

Formula

Multiplicative with a(p^e) = p^(2*e - 4) * (p^4 + e * (p^2 - 1)^2 - 1).
a(n) = Sum_{d|n} J_2(d) * J_2(n/d).
a(n) = Sum_{d|n} d^2 * tau(d) * A007427(n/d), where tau = A000005.
a(n) = Sum_{d|n} d^2 * A321322(n/d).
(1/tau(n)) * Sum_{d|n} a(d) * tau(n/d) = n^2.
Sum_{k=1..n} a(k) ~ ((3*log(n) + 6*gamma - 1)/(9*zeta(3)^2) - 2*zeta'(3) / (3*zeta(3)^3)) * n^3, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 14 2020

A341772 a(n) = Sum_{d|n} phi(d) * J_2(n/d).

Original entry on oeis.org

1, 4, 10, 17, 28, 40, 54, 70, 94, 112, 130, 170, 180, 216, 280, 284, 304, 376, 378, 476, 540, 520, 550, 700, 716, 720, 858, 918, 868, 1120, 990, 1144, 1300, 1216, 1512, 1598, 1404, 1512, 1800, 1960, 1720, 2160, 1890, 2210, 2632, 2200, 2254, 2840, 2682, 2864, 3040, 3060, 2860, 3432, 3640
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 19 2021

Keywords

Comments

Dirichlet convolution of Euler totient function phi (A000010) with Jordan function J_2 (A007434).

Links

Crossrefs

Cf. A000010, A000203, A001615, A002618, A007427, A007431, A007434, A008683, A023900, A029935, A064987, A069097, A321322, A322577, A338164.

Programs

  • Mathematica
    Jordan2[n_] := Sum[MoebiusMu[n/d] d^2, {d, Divisors[n]}]; a[n_] := Sum[EulerPhi[d] Jordan2[n/d], {d, Divisors[n]}]; Table[a[n], {n, 55}]
f[p_, e_] := p^(e-3)*(p-1)*(p^e*(p+1)^2-p); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 31 2024 *)
  • PARI
    J2(n) = sumdiv(n, d, d^2 * moebius(n/d)); \\ A007434
a(n) = sumdiv(n, d, eulerphi(d) * J2(n/d)); \\ Michel Marcus, Feb 20 2021

Formula

Dirichlet g.f.: zeta(s-1) * zeta(s-2) / zeta(s)^2.
a(n) = Sum_{k=1..n} J_2(gcd(n,k)).
a(n) = Sum_{d|n} psi(d) * phi(d) * phi(n/d).
a(n) = Sum_{d|n} d * phi(d) * A029935(n/d).
a(n) = Sum_{d|n} d * sigma(d) * A007427(n/d).
a(n) = Sum_{d|n} d * A321322(n/d).
a(n) = Sum_{d|n} d * A023900(d) * A338164(n/d).
a(n) = Sum_{d|n} d^2 * A007431(n/d).
a(n) = Sum_{d|n} mu(n/d) * A069097(d).
Sum_{k=1..n} a(k) ~ Pi^2 * n^3 / (18*zeta(3)^2). - Vaclav Kotesovec, Feb 20 2021
a(n) = Sum_{k=1..n} J_2(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021
a(n) = Sum_{1 <= i, j <= n} phi(gcd(i, j, n)). - Peter Bala, Jan 21 2024
Multiplicative with a(p^e) = p^(e-3)*(p-1)*(p^e*(p+1)^2-p). - Amiram Eldar, May 31 2024

A326828 a(n) = (1/2) * Sum_{d|n} mu(n/d) * phi(d) * (psi(d) + 1), where mu = A008683, phi = A000010 and psi = A001615.

Original entry on oeis.org

1, 1, 4, 5, 13, 7, 26, 19, 34, 23, 64, 32, 89, 47, 82, 74, 151, 64, 188, 105, 167, 119, 274, 127, 296, 167, 294, 214, 433, 161, 494, 292, 421, 287, 548, 290, 701, 359, 590, 417, 859, 329, 944, 540, 742, 527, 1126, 506, 1170, 576, 1012, 757, 1429, 576, 1382
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 20 2019

Keywords

Comments

Moebius transform applied twice to triangular numbers (A000217).

Crossrefs

Cf. A000010, A000217, A001615, A007427, A007431, A007438, A008683, A321322, A326826.

Programs

  • Maple
    with(numtheory):
b:= proc(n) option remember;
       add(mobius(n/d)*d*(d+1)/2, d=divisors(n))
    end:
a:= proc(n) option remember;
       add(mobius(n/d)*b(d), d=divisors(n))
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, Oct 20 2019
  • Mathematica
    Table[1/2 Sum[MoebiusMu[n/d] EulerPhi[d] (DirichletConvolve[j, MoebiusMu[j]^2, j, d] + 1), {d, Divisors[n]}], {n, 1, 55}]
Table[1/2 Sum[d (d + 1) DivisorSum[n/d, MoebiusMu[#] MoebiusMu[(n/d)/#] &], {d, Divisors[n]}], {n, 1, 55}]
nmax = 55; CoefficientList[Series[Sum[Sum[MoebiusMu[j] MoebiusMu[i] x^(i j)/(1 - x^(i j))^3, {j, 1, nmax}], {i, 1, nmax}], {x, 0, nmax}], x] // Rest

Formula

G.f.: Sum_{i>=1} Sum_{j>=1} mu(j) * mu(i) * x^(i*j) / (1 - x^(i*j))^3.
Dirichlet g.f.: (zeta(s-1) + zeta(s-2)) / (2 * zeta(s)^2).
a(n) = (1/2) * Sum_{d|n} mu(n/d) * (phi(d) + J_2(d)), where J_2 = A007434.
a(n) = (1/2) * Sum_{d|n} d * (d + 1) * A007427(n/d).
a(n) = Sum_{d|n} mu(n/d) * A007438(d).
Sum_{k=1..n} a(k) ~ n^3 / (6*zeta(3)^2). - Vaclav Kotesovec, Dec 11 2021
Showing 1-3 of 3 results.

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