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 1-3 of 3 results.

A342677 a(n) = Sum_{d|n} (n/d)^(n-d+1).

Original entry on oeis.org

1, 5, 28, 265, 3126, 46916, 823544, 16793633, 387422677, 10001953190, 285311670612, 8916464313700, 302875106592254, 11112103714568680, 437893891601739648, 18446779258148749825, 827240261886336764178, 39346424755299348744797, 1978419655660313589123980
Offset: 1

Views

Author

Seiichi Manyama, Mar 18 2021

Keywords

Crossrefs

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^(n - # + 1) &]; Array[a, 20] (* Amiram Eldar, Mar 18 2021 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^(n-d+1));
    
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (k*x)^k/(1-k^(k-1)*x^k)))

Formula

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

A356539 a(n) = Sum_{d|n} d * 3^(n-d).

Original entry on oeis.org

1, 5, 12, 49, 86, 492, 736, 3977, 8757, 34030, 59060, 384924, 531454, 2672528, 6672552, 26093113, 43046738, 261646137, 387420508, 2181624374, 4682526672, 17435870644, 31381059632, 204908769276, 299863458511, 1412168408630, 3392641222200, 13912336721584
Offset: 1

Views

Author

Seiichi Manyama, Aug 11 2022

Keywords

Crossrefs

Programs

  • Mathematica
    a[n_] := DivisorSum[n, # * 3^(n - #) &]; Array[a, 30] (* Amiram Eldar, Aug 11 2022 *)
  • PARI
    a(n) = sumdiv(n, d, d*3^(n-d));
    
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, k*x^k/(1-(3*x)^k)))

Formula

G.f.: Sum_{k>=1} k * x^k/(1 - (3 * x)^k).
If p is prime, a(p) = p + 3^(p-1).

A356529 a(n) = (n-1)! * Sum_{d|n} d^(n-d+1).

Original entry on oeis.org

1, 3, 8, 78, 144, 14400, 5760, 5851440, 88583040, 5859786240, 43545600, 24077414592000, 6706022400, 35948640894566400, 4395744249613516800, 263312496059348736000, 376610217984000, 5901087844517892009984000, 128047474114560000
Offset: 1

Views

Author

Seiichi Manyama, Aug 10 2022

Keywords

Crossrefs

Programs

  • Mathematica
    a[n_] := (n - 1)! * DivisorSum[n, #^(n - # + 1) &]; Array[a, 20] (* Amiram Eldar, Aug 10 2022 *)
  • PARI
    a(n) = (n-1)!*sumdiv(n, d, d^(n-d+1));
    
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, log(1-(k*x)^k)/k^k)))

Formula

a(n) = (n-1)! * A342675(n).
If p is prime, a(p) = (1+p) * (p-1)!.
E.g.f.: -Sum_{k>0} log(1 - (k * x)^k)/k^k.
Showing 1-3 of 3 results.