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.

A288391 Expansion of Product_{k>=1} 1/(1 - x^k)^(sigma_3(k)).

Original entry on oeis.org

1, 1, 10, 38, 156, 534, 2014, 6796, 23312, 76165, 247234, 780343, 2435903, 7453859, 22538336, 67130594, 197666509, 574876417, 1654464954, 4711217687, 13288453688, 37133349758, 102873771662, 282630567325, 770410193747, 2084205092693, 5598070811010
Offset: 0

Views

Author

Seiichi Manyama, Jun 08 2017

Keywords

Links

Crossrefs

Cf. A027848, A288392, A288415.
Product_{k>=1} 1/(1 - x^k)^sigma_m(k): A006171 (m=0), A061256 (m=1), A275585 (m=2), this sequence (m=3).

Programs

  • Magma
    m:=40; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-q^k)^DivisorSigma(3,k): k in [1..m]]) )); // G. C. Greubel, Oct 30 2018
  • Maple
    with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      d*sigma[3](d), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jun 08 2017
  • Mathematica
    nmax = 40; CoefficientList[Series[Product[1/(1-x^k)^DivisorSigma[3, k], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 23 2018 *)
  • PARI
    m=40; x='x+O('x^m); Vec(prod(k=1, m, 1/(1-x^k)^sigma(k,3))) \\ G. C. Greubel, Oct 30 2018

Formula

a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A027848(k)*a(n-k) for n > 0.
a(n) ~ exp((5*Pi)^(4/5) * Zeta(5)^(1/5) * n^(4/5) / (2^(8/5) * 3^(1/5)) - Zeta'(-3)/2) * Zeta(5)^(121/1200) / ((24*Pi)^(121/1200) * 5^(721/1200) * n^(721/1200)). - Vaclav Kotesovec, Mar 23 2018
G.f.: exp(Sum_{k>=1} sigma_4(k)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Oct 26 2018

A288420 a(n) = Sum_{d|n} d^4*A000593(n/d).

Original entry on oeis.org

1, 17, 85, 273, 631, 1445, 2409, 4369, 6898, 10727, 14653, 23205, 28575, 40953, 53635, 69905, 83539, 117266, 130341, 172263, 204765, 249101, 279865, 371365, 394406, 485775, 558778, 657657, 707311, 911795, 923553, 1118481, 1245505, 1420163, 1520079, 1883154
Offset: 1

Views

Author

Seiichi Manyama, Jun 09 2017

Keywords

Comments

Multiplicative because this sequence is the Dirichlet convolution of A000583 and A000593 which are both multiplicative. - Andrew Howroyd, Jul 20 2018

Links

Crossrefs

Cf. A000583, A013662, A013663, A027848, A288415.
Sum_{d|n} d^k*A000593(n/d): A288417 (k=0), A109386 (k=1), A288418 (k=2), A288419 (k=3), this sequence (k=4).

Programs

  • Mathematica
    f[p_, e_] :=  (p^(4*e+7) - (p^3+p^2+p+1)*p^(e+1) + p^2 + p + 1)/(p^7 - (p^3+p^2+p+1)*p + p^2 + p + 1); f[2, e_] := (16^(e+1)-1)/15; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 13 2022 *)

Formula

From Amiram Eldar, Nov 13 2022: (Start)
a(n) = A027848(n) for odd n.
Multiplicative with a(2^e) = (16^(e+1)-1)/15 and a(p^e) = (p^(4*e+7) - (p^3+p^2+p+1)*p^(e+1) + p^2 + p + 1)/(p^7 - (p^3+p^2+p+1)*p + p^2 + p + 1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^5, where c = Pi^4*zeta(5)/480 = (3/16)*zeta(4)*zeta(5) = 0.210429... . (End)

Extensions

Keyword:mult added by Andrew Howroyd, Jul 23 2018

A288392 Expansion of Product_{k>=1} (1 - x^k)^(sigma_3(k)).

Original entry on oeis.org

1, -1, -9, -19, -9, 163, 573, 1127, 109, -7198, -27159, -58611, -50378, 157532, 892986, 2431694, 4040909, 1605559, -16109148, -68261139, -167737209, -263590908, -109589779, 934422499, 3976197701, 9922490735, 16765911071, 13022553978, -33008232762
Offset: 0

Views

Author

Seiichi Manyama, Jun 08 2017

Keywords

Links

Crossrefs

Cf. A027848, A288391.
Product_{k>=1} (1 - x^k)^sigma_m(k): A288098 (m=0), A288385 (m=1), A288389 (m=2), this sequence (m=3).

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1-q^k)^DivisorSigma(3,k): k in [1..m]]) )); // G. C. Greubel, Oct 30 2018
  • Maple
    with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(
      d*sigma[3](d), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      -add(b(n-i)*a(i), i=0..n-1))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jun 08 2017
  • Mathematica
    nmax = 30; CoefficientList[Series[Product[(1-x^k)^DivisorSigma[3, k], {k, 1, nmax}], {x, 0, nmax}], x] (* G. C. Greubel, Oct 30 2018 *)
  • PARI
    m=30; x='x+O('x^m); Vec(prod(k=1, m, (1-x^k)^sigma(k,3))) \\ G. C. Greubel, Oct 30 2018

Formula

Convolution inverse of A288391.
a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A027848(k)*a(n-k) for n > 0.
G.f.: exp(-Sum_{k>=1} sigma_4(k)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Oct 29 2018

A320940 a(n) = Sum_{d|n} d*sigma_n(d).

Original entry on oeis.org

1, 11, 85, 1127, 15631, 287021, 5764809, 135007759, 3487020610, 100146496681, 3138428376733, 107032667155169, 3937376385699303, 155582338242604221, 6568408966322733475, 295154660699054931999, 14063084452067724991027, 708239400347943609329270
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 28 2018

Keywords

Examples

			a(6) = 1*sigma_6(1)+2*sigma_6(2)+3*sigma_6(3)+6*sigma_6(6) = 1+2*65+3*730+6*47450 = 287021.

Links

Crossrefs

Cf. A000203, A001001, A027847, A027848, A319647, A321141.

Programs

  • Magma
    [&+[d*DivisorSigma(n,d):d in Divisors(n)]:n in [1..18]]; // Marius A. Burtea, Feb 15 2020
  • Maple
    with(numtheory): seq(coeff(series(n*(-log(mul((1-x^k)^sigma[n](k),k=1..n))),x,n+1), x, n), n = 1 .. 20); # Muniru A Asiru, Oct 28 2018
  • Mathematica
    Table[Sum[d DivisorSigma[n, d], {d, Divisors[n]}] , {n, 18}]
Table[n SeriesCoefficient[-Log[Product[(1 - x^k)^DivisorSigma[n, k], {k, 1, n}]], {x, 0, n}], {n, 18}]
  • PARI
    a(n) = sumdiv(n, d, d*sigma(d, n)); \\ Michel Marcus, Oct 28 2018
  • Python
    from sympy import divisor_sigma, divisors
def A320940(n):
    return sum(divisor_sigma(d)*(n//d)**(n+1) for d in divisors(n,generator=True)) # Chai Wah Wu, Feb 15 2020

Formula

a(n) = n * [x^n] -log(Product_{k>=1} (1 - x^k)^sigma_n(k)).
a(n) = Sum_{d|n} d^(n+1)*sigma_1(n/d).
a(n) ~ n^(n+1). - Vaclav Kotesovec, Feb 16 2020

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

A322104 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} d*sigma_k(d).

Original entry on oeis.org

1, 1, 5, 1, 7, 7, 1, 11, 13, 17, 1, 19, 31, 35, 11, 1, 35, 85, 95, 31, 35, 1, 67, 247, 311, 131, 91, 15, 1, 131, 733, 1127, 631, 341, 57, 49, 1, 259, 2191, 4295, 3131, 1615, 351, 155, 34, 1, 515, 6565, 16775, 15631, 8645, 2409, 775, 130, 55, 1, 1027, 19687, 66311, 78131, 49111, 16815, 4991, 850, 217, 23
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 26 2018

Keywords

Examples

			Square array begins:
   1,   1,    1,     1,     1,      1,  ...
   5,   7,   11,    19,    35,     67,  ...
   7,  13,   31,    85,   247,    733,  ...
  17,  35,   95,   311,  1127,   4295,  ...
  11,  31,  131,   631,  3131,  15631,  ...
  35,  91,  341,  1615,  8645,  49111,  ...

Links

Crossrefs

Columns k=0..3 give A060640, A001001, A027847, A027848.
Cf. A109974, A320940 (diagonal), A321876, A322103.

Programs

  • Mathematica
    Table[Function[k, Sum[d DivisorSigma[k, d], {d, Divisors[n]}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[j 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+1)*sigma(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} j*sigma_k(j)*x^j/(1 - x^j).
L.g.f. of column k: -log(Product_{j>=1} (1 - x^j)^sigma_k(j)).
A(n,k) = Sum_{d|n} d^(k+1)*sigma_1(n/d).

A344787 a(n) = n * Sum_{d|n} sigma_d(d) / d, where sigma_k(n) is the sum of the k-th powers of the divisors of n.

Original entry on oeis.org

1, 7, 31, 287, 3131, 47527, 823551, 16843583, 387440266, 10009772937, 285311670623, 8918294639219, 302875106592267, 11112685050294387, 437893920912795941, 18447025553014982271, 827240261886336764195, 39346558271492953948522, 1978419655660313589123999
Offset: 1

Views

Author

Wesley Ivan Hurt, May 28 2021

Keywords

Comments

If p is prime, a(p) = p * Sum_{d|p} sigma_d(d) / d = p * (1 + (1^p + p^p)/p) = 1 + p + p^p.

Examples

			a(4) = 4 * Sum_{d|4} sigma_d(d) / d = 4 * ((1^1)/1 + (1^2 + 2^2)/2 + (1^4 + 2^4 + 4^4)/4) = 287.

Links

Crossrefs

Cf. A245466, A321141, A334874, A343781, A344434, A344480.
Cf. A001001, A007429, A027847, A027848, A060640.

Programs

  • Mathematica
    Table[n*Sum[DivisorSigma[k, k] (1 - Ceiling[n/k] + Floor[n/k])/k, {k, n}], {n, 20}]
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, sigma(k, k)*x^k/(1-x^k)^2)) \\ Seiichi Manyama, Dec 16 2022

Formula

G.f.: Sum_{k>=1} sigma_k(k) * x^k/(1 - x^k)^2. - Seiichi Manyama, Dec 16 2022
Showing 1-7 of 7 results.

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