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.

A354891 a(n) = n! * Sum_{d|n} d^(n - d) / d!.

Original entry on oeis.org

1, 3, 7, 73, 121, 9721, 5041, 1760641, 44452801, 562615201, 39916801, 3156125575681, 6227020801, 192873372531841, 222245415808416001, 14806216643368550401, 355687428096001, 34884164976924636172801, 121645100408832001
Offset: 1

Views

Author

Seiichi Manyama, Jun 10 2022

Keywords

Crossrefs

Programs

  • Mathematica
    a[n_] := n! * DivisorSum[n, #^(n - #)/#! &]; Array[a, 19] (* Amiram Eldar, Jun 10 2022 *)
  • PARI
    a(n) = n!*sumdiv(n, d, d^(n-d)/d!);
    
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, x^k/(k!*(1-(k*x)^k)))))

Formula

E.g.f.: Sum_{k>0} x^k/(k! * (1 - (k * x)^k)).
If p is prime, a(p) = 1 + p! = A038507(p).

A354897 a(n) = n! * Sum_{d|n} d^n / (d! * (n/d)!).

Original entry on oeis.org

1, 5, 28, 353, 3126, 94237, 823544, 72042497, 585825130, 157671732881, 285311670612, 790577855833537, 302875106592254, 5876819345289651137, 55890419425648520176, 73205730667453550166017, 827240261886336764178, 1474631675630757976051079425
Offset: 1

Views

Author

Seiichi Manyama, Jun 11 2022

Keywords

Crossrefs

Programs

  • Mathematica
    a[n_] := n! * DivisorSum[n, #^n/(#! * (n/#)!) &]; Array[a, 18] (* Amiram Eldar, Jun 11 2022 *)
  • PARI
    a(n) = n!*sumdiv(n, d, d^n/(d!*(n/d)!));
    
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, (exp((k*x)^k)-1)/k!)))

Formula

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

A354898 a(n) = n! * Sum_{d|n} d^(n - d) / (d! * (n/d)!).

Original entry on oeis.org

1, 2, 2, 26, 2, 2582, 2, 268802, 7348322, 51120722, 2, 299332756802, 2, 7157951760962, 18701679546950402, 613777679843328002, 2, 3250742570192384467202, 2, 29411516073133093829529602, 1146522800008167069616128002, 4017001663590220290585602, 2
Offset: 1

Views

Author

Seiichi Manyama, Jun 11 2022

Keywords

Crossrefs

Programs

  • Maple
    f:= proc(n) local d; n! * add(d^(n-d)/(d! * (n/d)!), d = numtheory:-divisors(n)) end proc:
    map(f, [$1..30]); # Robert Israel, Jul 10 2023
  • Mathematica
    a[n_] := n! * DivisorSum[n, #^(n - #)/(#! * (n/#)!) &]; Array[a, 23] (* Amiram Eldar, Jun 11 2022 *)
  • PARI
    a(n) = n!*sumdiv(n, d, d^(n-d)/(d!*(n/d)!));
    
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, (exp((k*x)^k)-1)/(k^k*k!))))

Formula

E.g.f.: Sum_{k>0} (exp((k * x)^k) - 1)/(k^k * k!).
If p is prime, a(p) = 2.
Showing 1-3 of 3 results.