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.

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

A386783 a(n) = n^4*sigma_2(n).

Original entry on oeis.org

0, 1, 80, 810, 5376, 16250, 64800, 120050, 348160, 597051, 1300000, 1786202, 4354560, 4855370, 9604000, 13162500, 22347776, 24221090, 47764080, 47176202, 87360000, 97240500, 142896160, 148315730, 282009600, 254296875, 388429600, 435781620, 645388800, 595530602, 1053000000
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 02 2025

Keywords

Links

Crossrefs

Cf. A280022, A280025, A386784, A386785, A386786, A386787, A386788.
Cf. A001157, A328259, A386745, A386746.

Programs

  • Magma
    [0] cat [n^4*DivisorSigma(2, n): n in [1..35]]; // Vincenzo Librandi, Aug 03 2025
  • Mathematica
    Table[n^4*DivisorSigma[2, n], {n, 0, 40}]
nmax = 40; CoefficientList[Series[Sum[k^4*x^k*(1 + 57*x^k + 302*x^(2*k) + 302*x^(3*k) + 57*x^(4*k) + x^(5*k)) / (1 - x^k)^7, {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Sum_{k>=1} k^4*x^k*(1 + 57*x^k + 302*x^(2*k) + 302*x^(3*k) + 57*x^(4*k) + x^(5*k)) / (1 - x^k)^7.
a(n) = n^4*A001157(n).
Dirichlet g.f.: zeta(s-4)*zeta(s-6). - 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

A386813 Coefficients in q-expansion of E_2^3 * E_4^2, where E_2 and E_4 are respectively the Eisenstein series A006352 and A004009.

Original entry on oeis.org

1, 408, 28872, -2685984, 24039336, 776610576, -657274464, -112765274688, -1315204139160, -9184174537416, -47705529895632, -201727238619744, -730623451715808, -2340991131399984, -6787572064867008, -18105120840067776, -44991518932447512, -105189400371536208, -233200610257765464
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 03 2025

Keywords

Crossrefs

Cf. A006352, A004009, A386781, A386785.

Programs

  • Mathematica
    terms = 20;
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}];
CoefficientList[Series[E2[x]^3*E4[x]^2, {x, 0, terms}], x]

A386814 Coefficients in q-expansion of E_2^4 * E_6, where E_2 and E_6 are respectively the Eisenstein series A006352 and A013973.

Original entry on oeis.org

1, -600, 34920, -157920, -23913240, 297457776, 3581091360, -13666238400, -458367407640, -4230394757880, -25457298127632, -118465178148000, -459399324219360, -1550209298287440, -4682236500918720, -12910757263315776, -32979872278342680, -78921341989665840, -178491991660958520
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 03 2025

Keywords

Crossrefs

Cf. A006352, A013973, A386785.

Programs

  • Mathematica
    terms = 20;
E2[x_] = 1 - 24*Sum[k*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[E2[x]^4*E6[x], {x, 0, terms}], x]
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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