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.

A355886 a(n) = n! * Sum_{k=1..n} floor(n/k)/k!.

Original entry on oeis.org

1, 5, 22, 125, 746, 5677, 44780, 420401, 4206970, 47543141, 562891352, 7573655905, 104684547566, 1596368400005, 25482043382476, 439969180782017, 7835163501390290, 151712475696833221, 3004182138648663200, 63854641556089628801, 1400563708969910620822
Offset: 1

Views

Author

Seiichi Manyama, Jul 20 2022

Keywords

Links

Crossrefs

Cf. A006218, A057625, A356010, A356458, A229837.
Cf. A356009, A356459.

Programs

  • Mathematica
    Table[n! * Sum[Floor[n/k]/k!, {k,1,n}], {n,1,25}] (* Vaclav Kotesovec, Aug 11 2025 *)
  • PARI
    a(n) = n!*sum(k=1, n, n\k/k!);
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, x^k/(k!*(1-x^k)))/(1-x)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, exp(x^k)-1)/(1-x)))
  • PARI
    a(n) = n!*sum(k=1, n, sumdiv(k, d, 1/d!)); \\ Seiichi Manyama, Aug 08 2022

Formula

E.g.f.: (1/(1-x)) * Sum_{k>0} x^k/(k! * (1 - x^k)).
E.g.f.: (1/(1-x)) * Sum_{k>0} (exp(x^k) - 1).
a(n) = n! * Sum_{k=1..n} Sum_{d|k} 1/d! = n! * Sum_{k=1..n} A057625(k)/k!. - Seiichi Manyama, Aug 08 2022
a(n) ~ A229837 * n! * n. - Vaclav Kotesovec, Aug 11 2025

A356297 a(n) = n! * Sum_{k=1..n} sigma_0(k)/k.

Original entry on oeis.org

1, 4, 16, 82, 458, 3228, 24036, 212448, 2032992, 21781440, 246853440, 3201742080, 42580650240, 621037186560, 9664270963200, 161166707251200, 2781679603046400, 52204357423411200, 1004687538456268800, 20823621371578368000, 447027656835852288000
Offset: 1

Views

Author

Seiichi Manyama, Aug 03 2022

Keywords

Crossrefs

Cf. A000005, A006218, A356010, A356298, A356323.

Programs

  • Mathematica
    Table[n! * Sum[DivisorSigma[0, k]/k, {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Aug 07 2022 *)
  • PARI
    a(n) = n!*sum(k=1, n, sigma(k, 0)/k);
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, log(1-x^k)/k)/(1-x)))

Formula

E.g.f.: -(1/(1-x)) * Sum_{k>0} log(1 - x^k)/k.
a(n) ~ n! * (log(n)^2/2 + 2*gamma*log(n) + gamma^2 - 2*sg1), where gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). - Vaclav Kotesovec, Aug 07 2022

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

A356335 Expansion of e.g.f. ( Product_{k>0} 1/(1-x^k) )^(1/(1-x)).

Original entry on oeis.org

1, 1, 6, 39, 332, 3290, 38994, 517986, 7762880, 128029464, 2311675560, 45188359920, 952047539112, 21452758881528, 515073388373712, 13114579450948920, 352881761400606720, 10000259978380933440, 297654582665846499264, 9280441162956638320704
Offset: 0

Views

Author

Seiichi Manyama, Aug 04 2022

Keywords

Crossrefs

Cf. A000041, A000203, A356010, A356336, A356337.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace((1/prod(k=1, N, 1-x^k))^(1/(1-x))))
  • PARI
    a356010(n) = n!*sum(k=1, n, sigma(k)/k);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a356010(j)*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A356010(k) * binomial(n-1,k-1) * a(n-k).

A356390 a(n) = n! * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d + 1) * d ) /k.

Original entry on oeis.org

1, 3, 17, 74, 514, 3564, 30708, 250704, 2780496, 29982240, 373350240, 4639870080, 67024333440, 988156834560, 16914631507200, 271941778483200, 4999620452198400, 94617104704819200, 1925772463506124800, 39245319872575488000, 902004581585737728000
Offset: 1

Views

Author

Seiichi Manyama, Aug 05 2022

Keywords

Links

Crossrefs

Cf. A356389, A356391.
Cf. A000593, A356010, A356393.

Programs

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

Formula

a(n) = n! * Sum_{k=1..n} A000593(k)/k.
E.g.f.: -(1/(1-x)) * Sum_{k>0} (-x)^k/(k * (1 - x^k)).
E.g.f.: (1/(1-x)) * Sum_{k>0} log(1 + x^k).
a(n) ~ n! * n * Pi^2/12. - Vaclav Kotesovec, Aug 07 2022

A356406 a(n) = n! * Sum_{k=1..n} Sum_{d|k} 1/(d * (k/d)^d).

Original entry on oeis.org

1, 4, 16, 79, 443, 2968, 22216, 189698, 1792402, 18745036, 213452996, 2653142952, 35448861576, 509724975264, 7824794618208, 128006170541328, 2217950478978576, 40686737647774368, 785852762719168992, 15974195890305405696, 340376906088298319616
Offset: 1

Views

Author

Seiichi Manyama, Aug 05 2022

Keywords

Crossrefs

Cf. A308345, A356009, A356010, A356407, A356408.

Programs

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

Formula

E.g.f.: -(1/(1-x)) * Sum_{k>0} log(1 - x^k/k).
a(n) = n! * Sum_{k=1..n} A308345(k)/k!.

A353992 a(n) = n! * Sum_{k=1..n} ( Sum_{d|k} d^(k/d + 1) )/k.

Original entry on oeis.org

1, 7, 41, 314, 2194, 22764, 195348, 2374224, 27940176, 384636960, 4673720160, 95522440320, 1323221996160, 23481816503040, 489968947641600, 10853692580505600, 190580382936115200, 5408424680491929600, 105077728210820198400, 3399507785578641408000
Offset: 1

Views

Author

Seiichi Manyama, Aug 06 2022

Keywords

Crossrefs

Cf. A078308, A353993, A356010, A356298, A356406.

Programs

  • Mathematica
    a[n_] := n! * Sum[DivisorSum[k, #^(k/# + 1) &]/k, {k, 1, n}]; Array[a, 20] (* Amiram Eldar, Aug 06 2022 *)
  • PARI
    a(n) = n!*sum(k=1, n, sumdiv(k, d, d^(k/d+1))/k);
  • PARI
    a(n) = n!*sum(k=1, n, sumdiv(k, d, (k/d)^d/d));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, log(1-k*x^k))/(1-x)))

Formula

a(n) = n! * Sum_{k=1..n} A078308(k)/k.
a(n) = n! * Sum_{k=1..n} Sum_{d|k} (k/d)^d / d.
E.g.f.: -(1/(1-x)) * Sum_{k>0} log(1 - k * x^k).

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

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 1-9 of 9 results.

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

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