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 1-5 of 5 results.

A386787 a(n) = n^4*sigma_7(n).

Original entry on oeis.org

0, 1, 2064, 177228, 4227328, 48828750, 365798592, 1977329144, 8657571840, 31395415077, 100782540000, 285311685252, 749200886784, 1792160422598, 4081207353216, 8653821705000, 17730707193856, 34271896391154, 64800136718928, 116490259028540, 206415142080000, 350438089532832
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 02 2025

Keywords

Links

Crossrefs

Cf. A280022, A386783, A280025, A386784, A386785, A386786, A386788.
Cf. A013955, A282060, A282753, A386781.
Cf. A386815, A386816, A282012, A386817, A282596, A386818, A282287.

Programs

  • Magma
    [0] cat [n^4*DivisorSigma(7, n): n in [1..35]]; // Vincenzo Librandi, Aug 03 2025
  • Mathematica
    Table[n^4*DivisorSigma[7, n], {n, 0, 30}]
(* or *)
nmax = 30; CoefficientList[Series[Sum[k^4*x^k*(1 + 2036*x^k + 152637*x^(2*k) + 2203488*x^(3*k) + 9738114*x^(4*k) + 15724248*x^(5*k) + 9738114*x^(6*k) + 2203488*x^(7*k) + 152637*x^(8*k) + 2036*x^(9*k) + x^(10*k))/(1 - x^k)^12, {k, 1, nmax}], {x, 0, nmax}], x]
(* or *)
terms = 30; E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}]; E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}]; E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}]; CoefficientList[Series[(33*E2[x]^4*E4[x]^2 + 110*E2[x]^2*E4[x]^3 + 13*E4[x]^4 - 132*E2[x]^3*E4[x]*E6[x] - 132*E2[x]*E4[x]^2*E6[x] + 88*E2[x]^2*E6[x]^2 + 20*E4[x]*E6[x]^2)/41472, {x, 0, terms}], x]

Formula

G.f.: Sum_{k>=1} k^4*x^k*(1 + 2036*x^k + 152637*x^(2*k) + 2203488*x^(3*k) + 9738114*x^(4*k) + 15724248*x^(5*k) + 9738114*x^(6*k) + 2203488*x^(7*k) + 152637*x^(8*k) + 2036*x^(9*k) + x^(10*k))/(1 - x^k)^12.
a(n) = (33*A386815(n) + 110*A386816(n) + 13*A282012(n) - 132*A386817(n) - 132*A282596(n) + 88*A386818(n) + 20*A282287(n))/41472.
a(n) = n^4*A013955(n).
Dirichlet g.f.: zeta(s-4)*zeta(s-11). - R. J. Mathar, Aug 03 2025

A386785 a(n) = n^4*sigma_5(n).

Original entry on oeis.org

0, 1, 528, 19764, 270592, 1953750, 10435392, 40356008, 138547200, 389021373, 1031580000, 2357962332, 5347980288, 10604527934, 21307972224, 38613915000, 70936231936, 118587960018, 205403284944, 322687828100, 528669120000, 797596142112, 1245004111296, 1801152941304, 2738246860800
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 02 2025

Keywords

Links

Crossrefs

Cf. A280022, A386783, A280025, A386784, A386786, A386787, A386788.
Cf. A001160, A282050, A282751, A282781.
Cf. A386813, A282549, A386814, A282792, A058550, A282576.

Programs

  • Magma
    [0] cat [n^4*DivisorSigma(5, n): n in [1..35]]; // Vincenzo Librandi, Aug 04 2025
  • Mathematica
    Table[n^4*DivisorSigma[5, n], {n, 0, 30}]
(* or *)
nmax = 30; CoefficientList[Series[Sum[k^4*x^k*(1 + 502*x^k + 14608*x^(2*k) + 88234*x^(3*k) + 156190*x^(4*k) + 88234*x^(5*k) + 14608*x^(6*k) + 502*x^(7*k) + x^(8*k))/(1 - x^k)^10, {k, 1, nmax}], {x, 0, nmax}], x]
(* or *)
terms = 30; E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}]; E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}]; E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}]; CoefficientList[Series[(4*E2[x]^3*E4[x]^2 + 2*E2[x]*E4[x]^3 - E2[x]^4*E6[x] - 6*E2[x]^2*E4[x]*E6[x] - E4[x]^2*E6[x] + 2*E2[x]*E6[x]^2)/3456, {x, 0, terms}], x]

Formula

G.f.: Sum_{k>=1} k^4*x^k*(1 + 502*x^k + 14608*x^(2*k) + 88234*x^(3*k) + 156190*x^(4*k) + 88234*x^(5*k) + 14608*x^(6*k) + 502*x^(7*k) + x^(8*k))/(1 - x^k)^10.
a(n) = (4*A386813(n) + 2*A282549(n) - A386814(n) - 6*A282792(n) - A058550(n) + 2*A282576(n))/3456.
a(n) = n^4*A001160(n).
Dirichlet g.f.: zeta(s-4)*zeta(s-9). - R. J. Mathar, Aug 03 2025

A386784 a(n) = n^4*sigma_4(n).

Original entry on oeis.org

0, 1, 272, 6642, 69888, 391250, 1806624, 5767202, 17895424, 43584723, 106420000, 214373522, 464196096, 815759282, 1568678944, 2598682500, 4581294080, 6975840962, 11855044656, 16983693362, 27343680000, 38305755684, 58309597984, 78311265122, 118861406208, 152832421875
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 02 2025

Keywords

Links

Crossrefs

Cf. A280022, A386783, A280025, A386785, A386786, A386787, A386788.
Cf. A001159, A386749, A386747, A386748.

Programs

  • Magma
    [0] cat [n^4*DivisorSigma(4, n): n in [1..35]]; // Vincenzo Librandi, Aug 03 2025
  • Mathematica
    Table[n^4*DivisorSigma[4, n], {n, 0, 40}]
nmax = 40; CoefficientList[Series[Sum[k^4*x^k*(1 + 247*x^k + 4293*x^(2*k) + 15619*x^(3*k) + 15619*x^(4*k) + 4293*x^(5*k) + 247*x^(6*k) + x^(7*k))/(1 - x^k)^9, {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Sum_{k>=1} k^4*x^k*(1 + 247*x^k + 4293*x^(2*k) + 15619*x^(3*k) + 15619*x^(4*k) + 4293*x^(5*k) + 247*x^(6*k) + x^(7*k))/(1 - x^k)^9.
a(n) = n^4*A001159(n).
Dirichlet g.f.: zeta(s-4)*zeta(s-8). - R. J. Mathar, Aug 03 2025

A386786 a(n) = n^4*sigma_6(n).

Original entry on oeis.org

0, 1, 1040, 59130, 1065216, 9766250, 61495200, 282477650, 1090785280, 3491573931, 10156900000, 25937439242, 62986222080, 137858520410, 293776756000, 577478362500, 1116964192256, 2015993983970, 3631236888240, 6131066388122, 10403165760000, 16702903444500, 26974936811680
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 02 2025

Keywords

Links

Crossrefs

Cf. A280022, A386783, A280025, A386784, A386785, A386787, A386788.
Cf. A013954, A386750, A386777, A386780.

Programs

  • Magma
    [0] cat [n^4*DivisorSigma(6, n): n in [1..35]]; // Vincenzo Librandi, Aug 03 2025
  • Mathematica
    Table[n^4*DivisorSigma[6, n], {n, 0, 30}]
nmax = 30; CoefficientList[Series[Sum[k^4*x^k*(1 + 1013*x^k + 47840*x^(2*k) + 455192*x^(3*k) + 1310354*x^(4*k) + 1310354*x^(5*k) + 455192*x^(6*k) + 47840*x^(7*k) + 1013*x^(8*k) + x^(9*k))/(1 - x^k)^11, {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Sum_{k>=1} k^4*x^k*(1 + 1013*x^k + 47840*x^(2*k) + 455192*x^(3*k) + 1310354*x^(4*k) + 1310354*x^(5*k) + 455192*x^(6*k) + 47840*x^(7*k) + 1013*x^(8*k) + x^(9*k))/(1 - x^k)^11.
a(n) = n^4*A013954(n).
Dirichlet g.f.: zeta(s-4)*zeta(s-10). - R. J. Mathar, Aug 03 2025

A386788 a(n) = n^4*sigma_8(n).

Original entry on oeis.org

0, 1, 4112, 531522, 16843008, 244141250, 2185618464, 13841289602, 68988964864, 282472589763, 1003908820000, 3138428391362, 8952429298176, 23298085151042, 56915382843424, 129766445482500, 282578800148480, 582622237313282, 1161527289105456, 2213314919196482, 4112073026880000
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 02 2025

Keywords

Links

Crossrefs

Cf. A280022, A386783, A280025, A386784, A386785, A386786, A386787.
Cf. A013956, A386751, A386778, A386782.

Programs

  • Magma
    [0] cat [n^4*DivisorSigma(8, n): n in [1..35]]; // Vincenzo Librandi, Aug 03 2025
  • Mathematica
    Table[n^4*DivisorSigma[8, n], {n, 0, 30}]
nmax = 30; CoefficientList[Series[Sum[k^4*x^k*(1 + 4083*x^k + 478271*x^(2*k) + 10187685*x^(3*k) + 66318474*x^(4*k) + 162512286*x^(5*k) + 162512286*x^(6*k) + 66318474*x^(7*k) + 10187685*x^(8*k) + 478271*x^(9*k) + 4083*x^(10*k) + x^(11*k))/(1 - x^k)^13, {k, 1, nmax}], {x, 0, nmax}], x]
  • PARI
    a(n) = if (n, n^4*sigma(n,8), 0); \\ Michel Marcus, Aug 03 2025

Formula

G.f.: Sum_{k>=1} k^4*x^k*(1 + 4083*x^k + 478271*x^(2*k) + 10187685*x^(3*k) + 66318474*x^(4*k) + 162512286*x^(5*k) + 162512286*x^(6*k) + 66318474*x^(7*k) + 10187685*x^(8*k) + 478271*x^(9*k) + 4083*x^(10*k) + x^(11*k))/(1 - x^k)^13.
a(n) = n^4*A013956(n).
Dirichlet g.f.: zeta(s-4)*zeta(s-12). - R. J. Mathar, Aug 03 2025
Showing 1-5 of 5 results.

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