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 1-8 of 8 results.

A338661 a(n) = Sum_{d|n} d^n * binomial(d+n/d-2, d-1).

Original entry on oeis.org

1, 5, 28, 289, 3126, 49036, 823544, 17040385, 387538588, 10048833246, 285311670612, 8929334253419, 302875106592254, 11116754387182648, 437894348359764856, 18448995959423107073, 827240261886336764178, 39347761059781438793815, 1978419655660313589123980
Offset: 1

Views

Author

Seiichi Manyama, Apr 22 2021

Keywords

Links

Crossrefs

Cf. A023887, A157019, A157020, A324158, A324159, A339481, A339482, A339712, A343573.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, #^n * Binomial[# + n/# - 2, #-1] &]; Array[a, 20] (* Amiram Eldar, Apr 22 2021 *)
  • PARI
    a(n) = sumdiv(n, d, d^n*binomial(d+n/d-2, d-1));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (k*x/(1-(k*x)^k))^k))

Formula

G.f.: Sum_{k >= 1} (k * x/(1 - (k * x)^k))^k.
If p is prime, a(p) = 1 + p^p.
a(n) ~ n^n. - Vaclav Kotesovec, Aug 04 2025

A339481 a(n) = Sum_{d|n} d^(n-d) * binomial(d+n/d-2, d-1).

Original entry on oeis.org

1, 2, 2, 10, 2, 131, 2, 1282, 4376, 16907, 2, 1138272, 2, 5793475, 154455992, 469893122, 2, 49501130330, 2, 1318441711177, 19001093813466, 3138439911059, 2, 15989399214596398, 6675720214843752, 3937376603803099, 6754271297694102092, 47097064577536888014, 2
Offset: 1

Views

Author

Seiichi Manyama, Apr 24 2021

Keywords

Crossrefs

Cf. A157019, A157020, A324158, A324159, A338661, A339482, A339712, A343573.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, #^(n - #) * Binomial[# + n/# - 2, # - 1] &]; Array[a, 30] (* Amiram Eldar, Apr 25 2021 *)
  • PARI
    a(n) = sumdiv(n, d, d^(n-d)*binomial(d+n/d-2, d-1));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (x/(1-(k*x)^k))^k))

Formula

G.f.: Sum_{k >= 1} (x/(1 - (k * x)^k))^k.
If p is prime, a(p) = 2.

A339712 a(n) = Sum_{d|n} d^(d+n/d-1) * binomial(d+n/d-2, d-1).

Original entry on oeis.org

1, 5, 28, 273, 3126, 46948, 823544, 16781441, 387421948, 10000078446, 285311670612, 8916102176891, 302875106592254, 11112006865913416, 437893890382064056, 18446744074783625217, 827240261886336764178, 39346408075327954053967, 1978419655660313589123980
Offset: 1

Views

Author

Seiichi Manyama, Apr 24 2021

Keywords

Crossrefs

Cf. A157019, A157020, A324158, A324159, A338661, A339481, A339482, A343573.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, #^(# + n/# - 1) * Binomial[# + n/# - 2, # - 1] &]; Array[a, 20] (* Amiram Eldar, Apr 25 2021 *)
  • PARI
    a(n) = sumdiv(n, d, d^(d+n/d-1)*binomial(d+n/d-2, d-1));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (k*x/(1-k*x^k))^k))

Formula

G.f.: Sum_{k >= 1} (k * x/(1 - k * x^k))^k.
If p is prime, a(p) = 1 + p^p.

A343574 a(n) = Sum_{d|n} d^d * binomial(d+n/d-1, d).

Original entry on oeis.org

1, 6, 30, 272, 3130, 46794, 823550, 16778544, 387420768, 10000018820, 285311670622, 8916100779324, 302875106592266, 11112006832146486, 437893890380925960, 18446744073860555680, 827240261886336764194, 39346408075300413088392
Offset: 1

Views

Author

Seiichi Manyama, Apr 20 2021

Keywords

Crossrefs

Cf. A081543, A343568, A343573.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, #^#*Binomial[# + n/# - 1, #] &]; Array[a, 20] (* Amiram Eldar, Apr 20 2021 *)
  • PARI
    a(n) = sumdiv(n, d, d^d*binomial(d+n/d-1, d));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (k*x)^k/(1-x^k)^(k+1)))

Formula

G.f.: Sum_{k >= 1} (k * x)^k/(1 - x^k)^(k+1).
If p is prime, a(p) = p + p^p.

A339482 a(n) = Sum_{d|n} d^(n-d+1) * binomial(d+n/d-2, d-1).

Original entry on oeis.org

1, 3, 4, 21, 6, 346, 8, 4617, 13132, 80696, 12, 4903847, 14, 40410966, 756336736, 2416181265, 18, 306560794753, 20, 6941876836216, 132964265599502, 34522735212626, 24, 116720277621236637, 33378601074218776, 51185893450298400, 60788365423272068968
Offset: 1

Views

Author

Seiichi Manyama, Apr 24 2021

Keywords

Crossrefs

Cf. A157019, A157020, A324158, A324159, A338661, A339481, A339712, A343573.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, #^(n - # + 1) * Binomial[# + n/# - 2, # - 1] &]; Array[a, 30] (* Amiram Eldar, Apr 25 2021 *)
  • PARI
    a(n) = sumdiv(n, d, d^(n-d+1)*binomial(d+n/d-2, d-1));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, k*(x/(1-(k*x)^k))^k))

Formula

G.f.: Sum_{k >= 1} k * (x/(1 - (k * x)^k))^k.
If p is prime, a(p) = 1 + p.

A358389 a(n) = n * Sum_{d|n} (d + n/d - 2)!/d!.

Original entry on oeis.org

1, 3, 7, 29, 121, 745, 5041, 40425, 362917, 3629411, 39916801, 479006233, 6227020801, 87178326495, 1307674369891, 20922790211057, 355687428096001, 6402373709009185, 121645100408832001, 2432902008212933061, 51090942171709581289, 1124000727778046764823
Offset: 1

Views

Author

Seiichi Manyama, Nov 13 2022

Keywords

Links

Crossrefs

Cf. A038507, A343573.

Programs

  • Mathematica
    Table[n*DivisorSum[n, ((# + n/# - 2)!)/(#!) &], {n, 22}] (* Michael De Vlieger, Nov 13 2022 *)
  • PARI
    a(n) = n*sumdiv(n, d, (d+n/d-2)!/d!);
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, k!*(x/(1-x^k))^k))

Formula

G.f.: Sum_{k>0} k! * (x/(1 - x^k))^k.
If p is prime, a(p) = 1 + p!.

A366933 Expansion of Sum_{k>=1} k^4 * x^k/(1 - x^k)^4.

Original entry on oeis.org

1, 20, 91, 340, 660, 1836, 2485, 5560, 7536, 13280, 14927, 31360, 29016, 49924, 60390, 89776, 84490, 152496, 131651, 226520, 227066, 299420, 282141, 514080, 415425, 581776, 614070, 850864, 711776, 1226520, 928977, 1442400, 1362042, 1693064, 1644930, 2609076
Offset: 1

Views

Author

Seiichi Manyama, Oct 29 2023

Keywords

Crossrefs

Cf. A064987, A366135, A366934.
Cf. A059358, A343544.
Cf. A343573.

Programs

  • PARI
    a(n) = sumdiv(n, d, d^4*binomial(n/d+2, 3));

Formula

a(n) = Sum_{d|n} d^4 * binomial(n/d+2,3).

A366934 Expansion of Sum_{k>=1} k^5 * x^k/(1 - x^k)^5.

Original entry on oeis.org

1, 37, 258, 1219, 3195, 9597, 17017, 39338, 63189, 118580, 162052, 316974, 373113, 630959, 826320, 1262692, 1424702, 2353896, 2483414, 3912790, 4397862, 6003569, 6451293, 10240908, 10004850, 13819832, 15382332, 20810398, 20547109, 30847530, 28675527, 40458504, 41853306
Offset: 1

Views

Author

Seiichi Manyama, Oct 29 2023

Keywords

Crossrefs

Cf. A064987, A366135, A366933.
Cf. A073570, A343545.
Cf. A343573.

Programs

  • PARI
    a(n) = sumdiv(n, d, d^5*binomial(n/d+3, 4));

Formula

a(n) = Sum_{d|n} d^5 * binomial(n/d+3,4).
Showing 1-8 of 8 results.

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

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