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 1-4 of 4 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

A386781 a(n) = n^3*sigma_7(n).

Original entry on oeis.org

0, 1, 1032, 59076, 1056832, 9765750, 60966432, 282475592, 1082196480, 3488379453, 10078254000, 25937425932, 62433407232, 137858494046, 291514810944, 576921447000, 1108169199616, 2015993905362, 3600007595496, 6131066264660, 10320757104000, 16687528072992, 26767423561824
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 02 2025

Keywords

Links

Crossrefs

Cf. A282211, A386746, A282213, A386748, A282781, A386780, A386782.
Cf. A013955, A282060, A282753.
Cf. A386813, A282549, A282792, A058550, A282576.

Programs

  • Magma
    [0] cat [n^3*DivisorSigma(7, n): n in [1..35]]; // Vincenzo Librandi, Aug 04 2025
  • Mathematica
    Table[n^3*DivisorSigma[7, n], {n, 0, 30}]
(* or *)
nmax = 30; CoefficientList[Series[Sum[k^10*x^k*(x^(2*k) + 4*x^k + 1)/(1 - x^k)^4, {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[(3*E2[x]^3*E4[x]^2 + 5*E2[x]*E4[x]^3 - 9*E2[x]^2*E4[x]*E6[x] - 3*E4[x]^2*E6[x] + 4*E2[x]*E6[x]^2)/3456, {x, 0, terms}], x]

Formula

G.f.: Sum_{k>=1} k^10*x^k*(x^(2*k) + 4*x^k + 1)/(1 - x^k)^4.
a(n) = (3*A386813(n) + 5*A282549(n) - 9*A282792(n) - 3*A058550(n) + 4*A282576(n))/3456.
a(n) = n^3*A013955(n).
Dirichlet g.f.: zeta(s-3)*zeta(s-10). - R. J. Mathar, Aug 03 2025

A386780 a(n) = n^3*sigma_6(n).

Original entry on oeis.org

0, 1, 520, 19710, 266304, 1953250, 10249200, 40353950, 136348160, 387952659, 1015690000, 2357949022, 5248851840, 10604501570, 20984054000, 38498557500, 69810262016, 118587881410, 201735382680, 322687704638, 520158288000, 795376354500, 1226133491440, 1801152673630
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 02 2025

Keywords

Links

Crossrefs

Cf. A282211, A386746, A282213, A386748, A282781, A386781, A386782.
Cf. A013954, A386750, A386777.

Programs

  • Magma
    [0] cat [n^3*DivisorSigma(6, n): n in [1..35]]; // Vincenzo Librandi, Aug 04 2025
  • Mathematica
    Table[n^3*DivisorSigma[6, n], {n, 0, 30}]
nmax = 30; CoefficientList[Series[Sum[k^9*x^k*(x^(2*k) + 4*x^k + 1)/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Sum_{k>=1} k^9*x^k*(x^(2*k) + 4*x^k + 1)/(1 - x^k)^4.
a(n) = n^3*A013954(n).
Dirichlet g.f.: zeta(s-3)*zeta(s-9). - R. J. Mathar, Aug 03 2025

A386782 a(n) = n^3*sigma_8(n).

Original entry on oeis.org

0, 1, 2056, 177174, 4210752, 48828250, 364269744, 1977327086, 8623620608, 31385843307, 100390882000, 285311671942, 746035774848, 1792160396234, 4065384488816, 8651096365500, 17661175009280, 34271896312546, 64529293839192, 116490258905078, 205603651344000, 350330949134964
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 02 2025

Keywords

Links

Crossrefs

Cf. A282211, A386746, A282213, A386748, A282781, A386780, A386781.
Cf. A013956, A386751, A386778.

Programs

  • Magma
    [0] cat [n^3*DivisorSigma(8, n): n in [1..35]]; // Vincenzo Librandi, Aug 04 2025
  • Mathematica
    Table[n^3*DivisorSigma[8, n], {n, 0, 30}]
nmax = 30; CoefficientList[Series[Sum[k^11*x^k*(x^(2*k) + 4*x^k + 1)/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Sum_{k>=1} k^11*x^k*(x^(2*k) + 4*x^k + 1)/(1 - x^k)^4.
a(n) = n^3*A013956(n).
Dirichlet g.f.: zeta(s-3)*zeta(s-11). - R. J. Mathar, Aug 03 2025
Showing 1-4 of 4 results.

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