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.

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

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

A386815 Coefficients in q-expansion of E_2^4 * E_4^2, where E_2 and E_4 are respectively the Eisenstein series A006352 and A004009.

Original entry on oeis.org

1, 384, 19008, -3408384, 86384832, 390216960, -20773815552, -154767455232, 1360271378880, 30429758560128, 278226995437440, 1749537534970368, 8664534035259648, 36062711146189056, 131104383085776384, 427185615341306880, 1270776436150340544, 3499300888293305088, 9016032242401655616
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 03 2025

Keywords

Crossrefs

Cf. A006352, A004009, A386787.

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]^4*E4[x]^2, {x, 0, 20}], x]

A386816 Coefficients in q-expansion of E_2^2 * E_4^3, where E_2 and E_4 are respectively the Eisenstein series A006352 and A004009.

Original entry on oeis.org

1, 672, 145152, 8663424, -337036224, -6505531200, 40579467264, 1996981485312, 25931378854080, 210242562994464, 1273050737441280, 6245511315490944, 26057670474216192, 95466371280176064, 314217417062264832, 945050326572360960, 2631525623493208512, 6854684254893824832
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 03 2025

Keywords

Crossrefs

Cf. A006352, A004009, A386787.

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]^2*E4[x]^3, {x, 0, 20}], x]

A386817 Coefficients in q-expansion of E_2^3 * E_4 * E_6, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

1, -336, -114912, 4151616, 100931712, -2848456800, -37865826432, 222362076288, 7928555745600, 86986313152368, 620751040620480, 3392046804500928, 15293330001535488, 59435665658243616, 204976008706800384, 640351567531186560, 1840291945275505344, 4923361835292283488
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 03 2025

Keywords

Crossrefs

Cf. A006352, A004009, A013973, A386787.

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}];
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
CoefficientList[Series[E2[x]^3*E4[x]*E6[x], {x, 0, 20}], x]

A386818 Coefficients in q-expansion of E_2^2 * E_6^2, where E_2 and E_6 are respectively the Eisenstein series A006352 and A013973.

Original entry on oeis.org

1, -1056, 269568, 5490816, -301315008, -6705063360, 41022885888, 1997915006208, 25923296790720, 210257663162208, 1273067731422720, 6245405396604288, 26057761857270528, 95466552284986176, 314217210391363584, 945049912933328640, 2631525397984618944, 6854687219510589888
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 03 2025

Keywords

Crossrefs

Cf. A006352, A013973, A386787.

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]^2*E6[x]^2, {x, 0, 20}], x]
Showing 1-9 of 9 results.

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