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.

A356298 a(n) = n! * Sum_{k=1..n} sigma_2(k)/k.

Original entry on oeis.org

1, 7, 41, 290, 2074, 18444, 165108, 1749264, 19412496, 241299360, 3097006560, 45546606720, 673536159360, 10986261431040, 187460277177600, 3445281394329600, 64637392771123200, 1325310849663897600, 27498565425087590400, 616389533324974080000
Offset: 1

Views

Author

Seiichi Manyama, Aug 03 2022

Keywords

Crossrefs

Cf. A001157, A064602, A356010, A356297, A356323.

Programs

  • Mathematica
    Table[n! * Sum[DivisorSigma[2, k]/k, {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Aug 07 2022 *)
  • PARI
    a(n) = n!*sum(k=1, n, sigma(k, 2)/k);
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, x^k/(k*(1-x^k)^2))/(1-x)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, k*log(1-x^k))/(1-x)))

Formula

E.g.f.: (1/(1-x)) * Sum_{k>0} x^k/(k * (1 - x^k)^2).
E.g.f.: -(1/(1-x)) * Sum_{k>0} k * log(1 - x^k).
a(n) ~ n! * zeta(3) * n^2 / 2. - Vaclav Kotesovec, Aug 07 2022

A356336 Expansion of e.g.f. ( Product_{k>0} 1/(1-x^k)^(1/k) )^(1/(1-x)).

Original entry on oeis.org

1, 1, 5, 29, 219, 1949, 20587, 245237, 3289577, 48670973, 788572541, 13849348105, 262283664739, 5317530185889, 114939490137235, 2636612228192969, 63955437488072593, 1634890446576454297, 43920715897460109205, 1236660724225711901749, 36412086992371220561771
Offset: 0

Views

Author

Seiichi Manyama, Aug 04 2022

Keywords

Crossrefs

Cf. A000005, A028342, A356297, A356335, A356337.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace((1/prod(k=1, N, (1-x^k)^(1/k)))^(1/(1-x))))
  • PARI
    a356297(n) = n!*sum(k=1, n, sigma(k, 0)/k);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a356297(j)*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A356297(k) * binomial(n-1,k-1) * a(n-k).

A356389 a(n) = n! * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d + 1) ) /k.

Original entry on oeis.org

1, 2, 10, 34, 218, 1308, 10596, 74688, 793152, 7931520, 94504320, 1054218240, 14662840320, 205279764480, 3427909632000, 50923531008000, 907545606912000, 16335820924416000, 323185344975360000, 6220416698689536000, 140360358705186816000, 3087927891514109952000
Offset: 1

Views

Author

Seiichi Manyama, Aug 05 2022

Keywords

Links

Crossrefs

Cf. A356390, A356391.
Cf. A048272, A356297, A356392.

Programs

  • Mathematica
    Table[n! * Sum[Sum[-(-1)^(k/d), {d, Divisors[k]}]/k, {k, 1, n}], {n, 1, 25}] (* Vaclav Kotesovec, Aug 07 2022 *)
Table[n! * Sum[(2*DivisorSigma[0, 2*k] - 3*DivisorSigma[0, k])/k, {k, 1, n}], {n, 1, 25}] (* Vaclav Kotesovec, Aug 07 2022 *)
  • PARI
    a(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1))/k);
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, log(1+x^k)/k)/(1-x)))

Formula

a(n) = n! * Sum_{k=1..n} A048272(k)/k.
E.g.f.: (1/(1-x)) * Sum_{k>0} log(1 + x^k)/k.
a(n) ~ n! * log(2) * (log(n) + 2*gamma - log(2)/2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Aug 07 2022

A356323 a(n) = n! * Sum_{k=1..n} sigma_3(k)/k.

Original entry on oeis.org

1, 11, 89, 794, 6994, 72204, 753108, 8973264, 111281616, 1524322080, 21601104480, 340803192960, 5483287025280, 96044874750720, 1748238132614400, 34093033838438400, 682396164763084800, 14706429413353574400, 323342442475011993600, 7585740483060676608000
Offset: 1

Views

Author

Seiichi Manyama, Aug 03 2022

Keywords

Crossrefs

Cf. A001158, A064603, A356010, A356297, A356298.

Programs

  • Mathematica
    Table[n! * Sum[DivisorSigma[3, k]/k, {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Aug 07 2022 *)
  • PARI
    a(n) = n!*sum(k=1, n, sigma(k, 3)/k);
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, x^k*(1+x^k)/(k*(1-x^k)^3))/(1-x)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, k^2*log(1-x^k))/(1-x)))

Formula

E.g.f.: (1/(1-x)) * Sum_{k>0} x^k * (1 + x^k)/(k * (1 - x^k)^3).
E.g.f.: -(1/(1-x)) * Sum_{k>0} k^2 * log(1 - x^k).
a(n) ~ n! * Pi^4 * n^3 / 270. - Vaclav Kotesovec, Aug 07 2022

A356436 a(n) = n! * Sum_{k=1..n} ( Sum_{d|k} d^(k/d) )/k.

Original entry on oeis.org

1, 5, 23, 146, 874, 8124, 62628, 707664, 7860816, 103284000, 1179669600, 24454569600, 324615427200, 5740203974400, 119579523436800, 2688723275212800, 46084905896601600, 1383333631684300800, 26411386476116275200, 868104140064602112000
Offset: 1

Views

Author

Seiichi Manyama, Aug 07 2022

Keywords

Links

Crossrefs

Cf. A055225, A353992, A356297, A356437, A356439.

Programs

  • PARI
    a(n) = n!*sum(k=1, n, sumdiv(k, d, d^(k/d))/k);
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, log(1-k*x^k)/k)/(1-x)))

Formula

a(n) = n! * Sum_{k=1..n} A055225(k)/k.
E.g.f.: -(1/(1-x)) * Sum_{k>0} log(1 - k*x^k)/k.
a(n) ~ (n-1)! * 3^((n + 3 - mod(n,3))/3)/2. - Vaclav Kotesovec, Aug 07 2022

A356437 a(n) = n! * Sum_{k=1..n} sigma_k(k)/k.

Original entry on oeis.org

1, 7, 77, 1946, 84754, 6202524, 636369348, 89979720144, 16431405256656, 3796658174518560, 1077102230236529760, 368915006390671969920, 149873555740938949215360, 71297150722148582901815040, 39244301012876892023553235200
Offset: 1

Views

Author

Seiichi Manyama, Aug 07 2022

Keywords

Crossrefs

Cf. A023887, A356297, A356436, A356440.

Programs

  • Mathematica
    Table[n! * Sum[DivisorSigma[k, k]/k, {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Aug 07 2022 *)
  • PARI
    a(n) = n!*sum(k=1, n, sigma(k, k)/k);
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, log(1-(k*x)^k)/k)/(1-x)))

Formula

E.g.f.: -(1/(1-x)) * Sum_{k>0} log(1 - (k*x)^k)/k.
a(n) ~ n! * n^(n-1). - Vaclav Kotesovec, Aug 07 2022

A356485 a(n) = n! * Sum_{k=1..n} A000010(k)/k.

Original entry on oeis.org

1, 3, 13, 64, 416, 2736, 23472, 207936, 2113344, 22584960, 284722560, 3576337920, 52240412160, 768727895040, 12228344755200, 206114911027200, 3838718125670400, 71231050830643200, 1468632692485324800, 30345814652977152000, 666456931810639872000, 15172961921551171584000
Offset: 1

Views

Author

Vaclav Kotesovec, Aug 09 2022

Keywords

Links

Crossrefs

Cf. A000010, A002088, A011755, A356297, A356010, A356298, A356323.

Programs

  • Mathematica
    Table[n! * Sum[EulerPhi[k]/k, {k, 1, n}], {n, 1, 25}]
  • PARI
    a(n) = n!*sum(k=1, n, eulerphi(k)/k); \\ Michel Marcus, Aug 09 2022

Formula

a(n) ~ n! * 6*n/Pi^2.
Showing 1-7 of 7 results.

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