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 1-9 of 9 results.

A321141 a(n) = Sum_{d|n} sigma_n(d).

Original entry on oeis.org

1, 6, 29, 291, 3127, 48246, 823545, 16909060, 387459858, 10019533302, 285311670613, 8920489178073, 302875106592255, 11113363271736486, 437893951444713443, 18447307036548136965, 827240261886336764179, 39346708467688595378892, 1978419655660313589123981
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 28 2018

Keywords

Links

Crossrefs

Cf. A000005, A007429, A007433, A023887, A319194, A320940, A321140.

Programs

  • Magma
    [&+[DivisorSigma(n, d):d in Divisors(n)]:n in [1..20]]; // Vincenzo Librandi, Feb 16 2020
  • Maple
    with(numtheory): seq(coeff(series(add(sigma[n](k)*x^k/(1-x^k),k=1..n),x,n+1), x, n), n = 1 .. 20); # Muniru A Asiru, Oct 28 2018
  • Mathematica
    Table[Sum[DivisorSigma[n, d], {d, Divisors[n]}] , {n, 19}]
Table[SeriesCoefficient[Sum[DivisorSigma[n, k] x^k/(1 - x^k), {k, 1, n}], {x, 0, n}], {n, 19}]
  • PARI
    a(n) = sumdiv(n, d, sigma(d, n)); \\ Michel Marcus, Oct 28 2018
  • Python
    from sympy import divisor_sigma, divisors
def A321141(n):
    return sum(divisor_sigma(d,0)*(n//d)**n for d in divisors(n,generator=True)) # Chai Wah Wu, Feb 15 2020

Formula

a(n) = [x^n] Sum_{k>=1} sigma_n(k)*x^k/(1 - x^k).
a(n) = Sum_{d|n} d^n*tau(n/d).
a(n) ~ n^n. - Vaclav Kotesovec, Feb 16 2020

A134577 A127170 * A127648.

Original entry on oeis.org

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

Views

Author

Gary W. Adamson, Nov 02 2007

Keywords

Comments

Row sums = A007429: (1, 4, 5, 11, 7, 20, 9, 26, ...).
Left border = A000005: (1, 2, 2, 3, 2, 4, 2, ...).
A134577 * [1/1, 1/2, 1/3, ...] = A007425: (1, 3, 3, 6, 3, 9, 3, 10, ...).
A134577 * [1, 2, 3, ...] = A007433: (1, 6, 11, 27, 27, 66, ...).
A134577 * A000005 = A034761: (1, 6, 8, 23, 12, 48, ...).

Examples

			First few rows of the triangle:
  1;
  2, 2;
  2, 0, 3;
  3, 4, 0, 4;
  2, 0, 0, 0, 5
  4, 4, 6, 0, 0, 6;
  2, 0, 0, 0, 0, 0, 7;
  4, 6, 0, 8, 0, 0, 0, 8;
  ...

Crossrefs

Cf. A051731, A127170, A127648, A127093, A007429, A127170, A007433, A034761.

Formula

A127170 * A127648 = A051731^2 * A127648 = A051731 * A127093.

A321140 a(n) = Sum_{d|n} sigma_3(d).

Original entry on oeis.org

1, 10, 29, 83, 127, 290, 345, 668, 786, 1270, 1333, 2407, 2199, 3450, 3683, 5349, 4915, 7860, 6861, 10541, 10005, 13330, 12169, 19372, 15878, 21990, 21226, 28635, 24391, 36830, 29793, 42798, 38657, 49150, 43815, 65238, 50655, 68610, 63771, 84836, 68923, 100050, 79509, 110639, 99822
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 28 2018

Keywords

Comments

Inverse Möbius transform applied twice to cubes.

Links

Crossrefs

Cf. A001158, A007429, A007433, A027848.

Programs

  • Maple
    with(numtheory): seq(coeff(series(add(sigma[3](k)*x^k/(1-x^k),k=1..n),x,n+1), x, n), n = 1 .. 45); # Muniru A Asiru, Oct 28 2018
  • Mathematica
    Table[Sum[DivisorSigma[3, d], {d, Divisors[n]}] , {n, 45}]
nmax = 45; Rest[CoefficientList[Series[Sum[DivisorSigma[3, k] x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
f[p_, e_] := (p^3*(p^(3e+3) - e - 2) + e + 1)/(p^3 - 1)^2; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 25 2020 *)
  • PARI
    a(n) = sumdiv(n, d, sigma(d, 3)); \\ Michel Marcus, Oct 28 2018

Formula

G.f.: Sum_{k>=1} sigma_3(k)*x^k/(1 - x^k).
a(n) = Sum_{d|n} d^3*tau(n/d).
From Jianing Song, Oct 28 2018: (Start)
Multiplicative with a(p^e) = (p^3*(p^(3e+3) - e - 2) + e + 1)/(p^3 - 1)^2.
Dirichlet g.f.: zeta(s)^2*zeta(s-3). (End)
Sum_{k=1..n} a(k) ~ Pi^8 * n^4 / 32400. - Vaclav Kotesovec, Nov 08 2018

A321322 a(n) = Sum_{d|n} mu(n/d)*J_2(d), where J_2() is the Jordan function (A007434).

Original entry on oeis.org

1, 2, 7, 9, 23, 14, 47, 36, 64, 46, 119, 63, 167, 94, 161, 144, 287, 128, 359, 207, 329, 238, 527, 252, 576, 334, 576, 423, 839, 322, 959, 576, 833, 574, 1081, 576, 1367, 718, 1169, 828, 1679, 658, 1847, 1071, 1472, 1054, 2207, 1008, 2304, 1152, 2009, 1503, 2807, 1152, 2737
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 04 2018

Keywords

Comments

Möbius transform applied twice to squares.

Links

Crossrefs

Cf. A000010, A001615, A007427, A007431, A007433, A007434, A008683.

Programs

  • Mathematica
    Table[Sum[MoebiusMu[n/d] Sum[MoebiusMu[d/j] j^2, {j, Divisors[d]}], {d, Divisors[n]}], {n, 55}]
nmax = 55; Rest[CoefficientList[Series[Sum[DivisorSum[k, MoebiusMu[#] MoebiusMu[k/#] &] x^k (1 + x^k)/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x]]
f[p_, e_] := If[e == 1, p^2 - 2, (p^2 - 1)^2*p^(2*e - 4)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 26 2020 *)
  • PARI
    for(n=1, 100, print1(direuler(p=2, n, (1 - X)^2/(1 - p^2*X))[n], ", ")) \\ Vaclav Kotesovec, Dec 11 2021

Formula

G.f.: Sum_{k>=1} A007427(k)*x^k*(1 + x^k)/(1 - x^k)^3.
a(n) = Sum_{d|n} mu(n/d)*phi(d)*psi(d), where phi() is the Euler totient function (A000010) and psi() is the Dedekind psi function (A001615).
Multiplicative with a(p^e) = p^2 - 2 if e = 1 and (p^2 - 1)^2 * p^(2*e - 4) otherwise. - Amiram Eldar, Oct 26 2020
From Vaclav Kotesovec, Dec 11 2021: (Start)
Dirichlet g.f.: zeta(s-2) / zeta(s)^2.
Sum_{k=1..n} a(k) ~ n^3 / (3*zeta(3)^2). (End)
a(n) = Sum_{1 <= i, j <= n} mu(gcd(i, j, n)). - Peter Bala, Jan 21 2024

A322103 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} sigma_k(d).

Original entry on oeis.org

1, 1, 3, 1, 4, 3, 1, 6, 5, 6, 1, 10, 11, 11, 3, 1, 18, 29, 27, 7, 9, 1, 34, 83, 83, 27, 20, 3, 1, 66, 245, 291, 127, 66, 9, 10, 1, 130, 731, 1091, 627, 290, 51, 26, 6, 1, 258, 2189, 4227, 3127, 1494, 345, 112, 18, 9, 1, 514, 6563, 16643, 15627, 8330, 2403, 668, 102, 28, 3
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 26 2018

Keywords

Examples

			Square array begins:
  1,   1,   1,    1,     1,     1,  ...
  3,   4,   6,   10,    18,    34,  ...
  3,   5,  11,   29,    83,   245,  ...
  6,  11,  27,   83,   291,  1091,  ...
  3,   7,  27,  127,   627,  3127,  ...
  9,  20,  66,  290,  1494,  8330,  ...

Links

Crossrefs

Columns k=0..3 give A007425, A007429, A007433, A321140.
Cf. A109974, A321141 (diagonal), A356045.

Programs

  • Mathematica
    Table[Function[k, Sum[DivisorSigma[k, d], {d, Divisors[n]}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[DivisorSigma[k, j] x^j/(1 - x^j), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
  • PARI
    T(n,k)={sumdiv(n, d, d^k*numdiv(n/d))}
for(n=1, 10, for(k=0, 8, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 26 2018

Formula

G.f. of column k: Sum_{j>=1} sigma_k(j)*x^j/(1 - x^j).
A(n,k) = Sum_{d|n} d^k*tau(n/d).

A356042 a(n) = Sum_{k=1..n} sigma_2(k) * floor(n/k).

Original entry on oeis.org

1, 7, 18, 45, 72, 138, 189, 301, 403, 565, 688, 985, 1156, 1462, 1759, 2212, 2503, 3115, 3478, 4207, 4768, 5506, 6037, 7269, 7947, 8973, 9895, 11272, 12115, 13897, 14860, 16678, 18031, 19777, 21154, 23908, 25279, 27457, 29338, 32362, 34045, 37411, 39262, 42583
Offset: 1

Views

Author

Seiichi Manyama, Jul 24 2022

Keywords

Crossrefs

Partial sums of A007433.
Column k=2 of A356045.
Cf. A000005 (tau).

Programs

  • Mathematica
    Table[Sum[DivisorSigma[2, k]*Floor[n/k], {k, 1, n}], {n, 1, 50}] (* Vaclav Kotesovec, Aug 07 2022 *)
  • PARI
    a(n) = sum(k=1, n, sigma(k, 2)*(n\k));
  • PARI
    a(n) = sum(k=1, n, sumdiv(k, d, d^2*numdiv(k/d)));
  • PARI
    my(N=50, x='x+O('x^N)); Vec(sum(k=1, N, sigma(k, 2)*x^k/(1-x^k))/(1-x))

Formula

a(n) = Sum_{k=1..n} Sum_{d|k} d^2 * tau(k/d).
G.f.: (1/(1-x)) * Sum_{k>=1} sigma_2(k) * x^k/(1 - x^k).
a(n) ~ zeta(3)^2 * n^3 / 3. - Vaclav Kotesovec, Aug 07 2022

A326826 a(n) = (1/2) * Sum_{d|n} (sigma_1(d) + sigma_2(d)), where sigma_1 = A000203 and sigma_2 = A001157.

Original entry on oeis.org

1, 5, 8, 19, 17, 43, 30, 69, 60, 95, 68, 176, 93, 171, 166, 255, 155, 342, 192, 403, 303, 395, 278, 681, 358, 543, 490, 738, 437, 961, 498, 969, 709, 911, 720, 1476, 705, 1131, 978, 1603, 863, 1773, 948, 1732, 1440, 1643, 1130, 2634, 1284, 2110, 1648, 2391, 1433, 2882, 1706
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 20 2019

Keywords

Comments

Inverse Moebius transform applied twice to triangular numbers (A000217).

Crossrefs

Cf. A000005, A000203, A000217, A001157, A007429, A007433, A007437, A092346, A326828.

Programs

  • Magma
    [(1/2)*&+[DivisorSigma(1,d)+DivisorSigma(2,d):d in Divisors(n)]:n in [1..55]]; // Marius A. Burtea, Oct 20 2019
  • Maple
    with(numtheory):
a:= n-> add(d*(d+1)*tau(n/d), d=divisors(n))/2:
seq(a(n), n=1..60);  # Alois P. Heinz, Oct 20 2019
  • Mathematica
    Table[1/2 Sum[DivisorSigma[1, d] + DivisorSigma[2, d], {d, Divisors[n]}], {n, 1, 55}]
Table[1/2 Sum[d (d + 1) DivisorSigma[0, n/d], {d, Divisors[n]}], {n, 1, 55}]
nmax = 55; CoefficientList[Series[Sum[Sum[x^(i j)/(1 - x^(i j))^3, {j, 1, nmax}], {i, 1, nmax}], {x, 0, nmax}], x] // Rest
  • PARI
    a(n) = sumdiv(n, d, sigma(d)+sigma(d, 2))/2; \\ Michel Marcus, Oct 20 2019

Formula

G.f.: Sum_{i>=1} Sum_{j>=1} x^(i*j) / (1 - x^(i*j))^3.
G.f.: (1/2) * Sum_{i>=1} Sum_{j>=1} j * (j + 1) * x^(i*j) / (1 - x^(i*j)).
G.f.: (1/2) * Sum_{k>=1} (sigma_1(k) + sigma_2(k)) * x^k / (1 - x^k).
Dirichlet g.f.: zeta(s)^2 * (zeta(s-1) + zeta(s-2)) / 2.
a(n) = (1/2) * Sum_{d|n} d * (d + 1) * tau(n/d), where tau = A000005.
a(n) = Sum_{d|n} A007437(d).
Sum_{k=1..n} a(k) ~ zeta(3)^2 * n^3 / 6. - Vaclav Kotesovec, Dec 11 2021

A328490 Dirichlet g.f.: zeta(s)^2 * zeta(s-2)^2.

Original entry on oeis.org

1, 10, 20, 67, 52, 200, 100, 380, 282, 520, 244, 1340, 340, 1000, 1040, 1973, 580, 2820, 724, 3484, 2000, 2440, 1060, 7600, 1978, 3400, 3460, 6700, 1684, 10400, 1924, 9710, 4880, 5800, 5200, 18894, 2740, 7240, 6800, 19760, 3364, 20000, 3700, 16348, 14664
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 16 2019

Keywords

Comments

Dirichlet convolution of A001157 with itself.
Dirichlet convolution of A000005 with A034714.
Dirichlet convolution of A000290 with A007433.

Crossrefs

Cf. A000005, A000290, A001157, A001620, A007426, A007433, A034714, A034761.
Cf. A175705.

Programs

  • Magma
    [&+[DivisorSigma(2,d)*DivisorSigma(2, n div d):d in Divisors(n)]:n in [1..50]]; // Marius A. Burtea, Oct 16 2019
  • Mathematica
    Table[DivisorSum[n, DivisorSigma[2, #] DivisorSigma[2, n/#] &], {n, 1, 45}]
f[p_, e_] :=((e*(p^2-1)+p^2-3)*p^(2*e+4) + e*(p^2-1) + 3*p^2 - 1)/(p^2-1)^3 ; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 15 2023 *)
  • PARI
    for(n=1, 100, print1(direuler(p=2, n, 1/(1 - X)^2 / (1 - p^2*X)^2)[n], ", ")) \\ Vaclav Kotesovec, Sep 26 2020

Formula

a(n) = Sum_{d|n} sigma_2(d) * sigma_2(n/d), where sigma_2 = A001157.
a(n) = Sum_{d|n} d^2 * tau(d) * tau(n/d), where tau = A000005.
Sum_{k=1..n} a(k) ~ zeta(3) * n^3 * (zeta(3)*(log(n)/3 + 2*gamma/3 - 1/9) + 2*zeta'(3)/3), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 17 2019
Multiplicative with a(p^e) = ((e*(p^2-1)+p^2-3)*p^(2*e+4) + e*(p^2-1) + 3*p^2 - 1)/(p^2-1)^3. - Amiram Eldar, Sep 15 2023

A340850 Dirichlet g.f.: Sum_{n>0} a(n)/n^s = zeta(s) * zeta(s-2) / (zeta(s-1))^2.

Original entry on oeis.org

1, 1, 4, 5, 16, 4, 36, 21, 40, 16, 100, 20, 144, 36, 64, 85, 256, 40, 324, 80, 144, 100, 484, 84, 416, 144, 364, 180, 784, 64, 900, 341, 400, 256, 576, 200, 1296, 324, 576, 336, 1600, 144, 1764, 500, 640, 484, 2116, 340, 1800, 416, 1024, 720, 2704, 364, 1600, 756, 1296, 784
Offset: 1

Views

Author

Werner Schulte, Jan 24 2021

Keywords

Crossrefs

Cf. A000290, A001157, A002618, A007433, A018804, A023900, A060640, A328722.

Programs

  • Mathematica
    f[p_, e_] := (p^(2*e) - 1)*(p - 1)/(p + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 24 2021 *)

Formula

Multiplicative with a(1) = 1 and a(p^e) = (p^(2*e)-1) * (p-1) / (p+1) for prime p and e > 0.
Dirichlet convolution of A002618 and A023900.
Dirichlet convolution of A001157 and A328722.
Dirichlet inverse b(n) for n > 0 is multiplicative with b(1) = 1 and b(p^e) = -(p-1)^2 * e * p^(e-1) for prime p and e > 0.
Dirichlet convolution with A060640 equals A007433.
Dirichlet convolution with A018804 equals A000290.
Sum_{k=1..n} a(k) ~ c * n^3, where c = 12*zeta(3)/Pi^4 = 0.148083... . - Amiram Eldar, Oct 16 2022
Showing 1-9 of 9 results.

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