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 1-6 of 6 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

A282751 Expansion of phi_{7, 2}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

Original entry on oeis.org

0, 1, 132, 2196, 16912, 78150, 289872, 823592, 2164800, 4802733, 10315800, 19487292, 37138752, 62748686, 108714144, 171617400, 277094656, 410338962, 633960756, 893872100, 1321672800, 1808608032, 2572322544, 3404825976, 4753900800, 6105469375, 8282826552
Offset: 0

Views

Author

Seiichi Manyama, Feb 21 2017

Keywords

Comments

Multiplicative because A001160 is. - Andrew Howroyd, Jul 25 2018

Links

Crossrefs

Cf. A282097 (phi_{3, 2}), A282099 (phi_{5, 2}), this sequence (phi_{7, 2}), A282753 (phi_{9, 2}).
Cf. A282101 (E_2*E_4^2), A282595 (E_2^2*E_6), A013974 (E_4*E_6 = E_10).
Cf. A001160 (sigma_5(n)), A282050 (n*sigma_5(n)), this sequence (n^2*sigma_5(n)).
Cf. A013664.

Programs

  • Mathematica
    Table[n^2 * DivisorSigma[5, n], {n, 0, 30}] (* Amiram Eldar, Sep 06 2023 *)
nmax = 40; CoefficientList[Series[Sum[k^7*x^k*(1 + x^k)/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 02 2025 *)
  • PARI
    a(n) = if(n < 1, 0, n^2*sigma(n, 5)) \\ Andrew Howroyd, Jul 25 2018

Formula

a(n) = n^2*A001160(n) for n > 0.
a(n) = (2*A282101(n) - A282595(n) - A013974(n))/1728.
Sum_{k=1..n} a(k) ~ zeta(6) * n^8 / 8. - Amiram Eldar, Sep 06 2023
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^(2*e) * (p^(5*e+5)-1)/(p^5-1).
Dirichlet g.f.: zeta(s-2)*zeta(s-7). (End)
G.f.: Sum_{k>=1} k^7*x^k*(1 + x^k)/(1 - x^k)^3. - Vaclav Kotesovec, Aug 02 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

A386777 a(n) = n^2*sigma_6(n).

Original entry on oeis.org

0, 1, 260, 6570, 66576, 390650, 1708200, 5764850, 17043520, 43105851, 101569000, 214359002, 437404320, 815730890, 1498861000, 2566570500, 4363141376, 6975757730, 11207521260, 16983563402, 26007914400, 37875064500, 55733340520, 78310985810, 111975926400, 152597656875
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 02 2025

Keywords

Links

Crossrefs

Cf. A282097, A386745, A282099, A386747, A282751, A282753, A386778.
Cf. A013954, A386750.

Programs

  • Magma
    [0] cat [n^2*DivisorSigma(6, n): n in [1..35]]; // Vincenzo Librandi, Aug 04 2025
  • Mathematica
    Table[n^2*DivisorSigma[6, n], {n, 0, 30}]
nmax = 30; CoefficientList[Series[Sum[k^8*x^k*(1 + x^k)/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Sum_{k>=1} k^8*x^k*(1 + x^k)/(1 - x^k)^3.
a(n) = n^2*A013954(n).
Dirichlet g.f.: zeta(s-2)*zeta(s-8). - R. J. Mathar, Aug 03 2025

A386778 a(n) = n^2*sigma_8(n).

Original entry on oeis.org

0, 1, 1028, 59058, 1052688, 9765650, 60711624, 282475298, 1077952576, 3487315923, 10039088200, 25937424722, 62169647904, 137858492018, 290384606344, 576739757700, 1103823438080, 2015993900738, 3584960768844, 6131066258162, 10280182567200, 16682426149284, 26663672614216
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 02 2025

Keywords

Links

Crossrefs

Cf. A282097, A386745, A282099, A386747, A282751, A386777, A282753.
Cf. A013956, A386751.

Programs

  • Magma
    [0] cat [n^2*DivisorSigma(8, n): n in [1..35]]; // Vincenzo Librandi, Aug 04 2025
  • Mathematica
    Table[n^2*DivisorSigma[8, n], {n, 0, 30}]
nmax = 30; CoefficientList[Series[Sum[k^10*x^k*(1 + x^k)/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Sum_{k>=1} k^10*x^k*(1 + x^k)/(1 - x^k)^3.
a(n) = n^2*A013956(n).
Dirichlet g.f.: zeta(s-2)*zeta(s-10). - R. J. Mathar, Aug 03 2025

A280021 Expansion of phi_{11, 2}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

Original entry on oeis.org

0, 1, 2052, 177156, 4202512, 48828150, 363524112, 1977326792, 8606744640, 31382654013, 100195363800, 285311670732, 744500215872, 1792160394206, 4057474577184, 8650199741400, 17626613022976, 34271896307922, 64397206034676, 116490258898580, 205200886312800
Offset: 0

Views

Author

Seiichi Manyama, Feb 22 2017

Keywords

Comments

Multiplicative because A013957 is. - Andrew Howroyd, Jul 23 2018

Links

Crossrefs

Cf. A282097 (phi_{3, 2}), A282099 (phi_{5, 2}), A282751 (phi_{7, 2}), A282753 (phi_{9, 2}), this sequence (phi_{11, 2}).
Cf. A282549 (E_2*E_4^3), A282792 (E_2^2*E_4*E_6), A282576 (E_2*E_6^2), A058550 (E_4^2*E_6 = E_14).
Cf. A013957 (sigma_9(n)), A282254 (n*sigma_9(n)), this sequence (n^2*sigma_9(n)).
Cf. A013668 (zeta(10)).

Programs

  • Mathematica
    Table[If[n>0, n^2 * DivisorSigma[9, n], 0], {n, 0, 20}] (* Indranil Ghosh, Mar 12 2017 *)
  • PARI
    for(n=0, 20, print1(if(n==0, 0, n^2 * sigma(n, 9)),", ")) \\ Indranil Ghosh, Mar 12 2017

Formula

a(n) = n^2*A013957(n) for n > 0.
a(n) = (6*A282549(n) - 5*A282792(n) + 4*A282576(n) - 5*A058550(n))/1728.
Sum_{k=1..n} a(k) ~ zeta(10) * n^12 / 12. - Amiram Eldar, Sep 06 2023
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^(2*e) * (p^(9*e+9)-1)/(p^9-1).
Dirichlet g.f.: zeta(s-2)*zeta(s-11). (End)
Showing 1-6 of 6 results.

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