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 1-6 of 6 results.

A308594 a(n) = Sum_{d|n} d^(d+n).

Original entry on oeis.org

1, 17, 730, 65601, 9765626, 2176802276, 678223072850, 281474993488897, 150094635297530563, 100000000030517582222, 81402749386839761113322, 79496847203492408399442540, 91733330193268616658399616010, 123476695691248494372093865205800
Offset: 1

Views

Author

Seiichi Manyama, Jun 09 2019

Keywords

Links

Crossrefs

Cf. A062796, A294956, A308668.

Programs

  • Mathematica
    sp[n_]:=Module[{d=Divisors[n]},Table[d[[k]]^(d[[k]]+n),{k,Length[ d]}]] // Total; Array[sp,15] (* Harvey P. Dale, Jan 02 2020 *)
a[n_] := DivisorSum[n, #^(# + n) &]; Array[a, 14] (* Amiram Eldar, May 11 2021 *)
  • PARI
    a(n) = sumdiv(n, d, d^(d+n));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(x*deriv(-log(prod(k=1, N, (1-(k*x)^k)^(k^(k-1))))))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (k^2*x)^k/(1-(k*x)^k))) \\ Seiichi Manyama, Mar 16 2021
  • Python
    from sympy import divisors
def A308594(n): return sum(d**(d+n) for d in divisors(n,generator=True)) # Chai Wah Wu, Jun 19 2022

Formula

L.g.f.: -log(Product_{k>=1} (1 - (k*x)^k)^(k^(k-1))) = Sum_{k>=1} a(k)*x^k/k.
G.f.: Sum_{k>=1} (k^2 * x)^k/(1 - (k * x)^k). - Seiichi Manyama, Mar 16 2021

A359700 a(n) = Sum_{d|n} d^(d + n/d - 1).

Original entry on oeis.org

1, 5, 28, 265, 3126, 46754, 823544, 16778273, 387420733, 10000015690, 285311670612, 8916100733146, 302875106592254, 11112006831323074, 437893890380939688, 18446744073843786241, 827240261886336764178, 39346408075300026047027
Offset: 1

Views

Author

Seiichi Manyama, Jan 11 2023

Keywords

Links

Crossrefs

Cf. A014566, A055225, A087909, A294956, A353013, A353014.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, #^(# + n/# - 1) &]; Array[a, 20] (* Amiram Eldar, Aug 14 2023 *)
  • PARI
    a(n) = sumdiv(n, d, d^(d+n/d-1));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (k*x)^k/(1-k*x^k)))

Formula

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

A294957 Expansion of Product_{k>=1} 1/(1 - k*x^k)^(k^(k-1)).

Original entry on oeis.org

1, 1, 5, 32, 300, 3533, 51650, 894929, 17981196, 410826036, 10518152538, 298209605418, 9273131902539, 313758357802886, 11474239675400172, 450962279143408815, 18954601400362304902, 848385358833157331498, 40285279861744621069122
Offset: 0

Views

Author

Seiichi Manyama, Nov 12 2017

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = n^(n-1), g(n) = n.

Links

Crossrefs

Cf. A294956.

Programs

  • PARI
    N=66; x='x+O('x^N); Vec(1/prod(k=1, N, (1-k*x^k)^k^(k-1)))

Formula

a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} A294956(k)*a(n-k) for n > 0.

A308668 a(n) = Sum_{d|n} d^(n/d+n).

Original entry on oeis.org

1, 9, 82, 1089, 15626, 287010, 5764802, 135270401, 3487315843, 100244173394, 3138428376722, 107072686593858, 3937376385699290, 155601328490478978, 6568412173896940652, 295165920677390712833, 14063084452067724991010
Offset: 1

Views

Author

Seiichi Manyama, Jun 16 2019

Keywords

Links

Crossrefs

Diagonal of A308502.
Cf. A152211, A294956, A308594.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, #^(n/# + n) &]; Array[a, 20] (* Amiram Eldar, Mar 17 2021 *)
  • PARI
    a(n) = sumdiv(n,d,d^(n/d+n));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(x*deriv(-log(prod(k=1, N, (1-k*(k*x)^k)^(1/k)))))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, k^(k+1)*x^k/(1-k^(k+1)*x^k))) \\ Seiichi Manyama, Mar 17 2021
  • Python
    from sympy import divisors
def A308668(n): return sum(d**(n//d+n) for d in divisors(n,generator=True)) # Chai Wah Wu, Jun 19 2022

Formula

L.g.f.: -log(Product_{k>=1} (1 - k*(k*x)^k)^(1/k)) = Sum_{k>=1} a(k)*x^k/k.
G.f.: Sum_{k>=1} k^(k+1) * x^k/(1 - k^(k+1) * x^k). - Seiichi Manyama, Mar 17 2021
a(n) ~ n^(n+1). - Vaclav Kotesovec, Aug 30 2025

A295234 Expansion of Product_{k>=1} (1 - k*x^k)^(k^(k-1)).

Original entry on oeis.org

1, -1, -4, -23, -225, -2765, -42291, -758931, -15672042, -365632740, -9512462314, -273071185192, -8574979449941, -292421476560437, -10762598186760785, -425244979326332068, -17953805056325313497, -806668085786772161511
Offset: 0

Views

Author

Seiichi Manyama, Nov 18 2017

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -n^(n-1), g(n) = n.

Links

Crossrefs

Cf. A266964, A294956, A294957.

Programs

  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-k*x^k)^k^(k-1)))

Formula

Convolution inverse of A294957.
a(0) = 1 and a(n) = -(1/n) * Sum_{k=1..n} A294956(k)*a(n-k) for n > 0.

A359701 a(n) = Sum_{d|n} d^(d + n/d - 2).

Original entry on oeis.org

1, 3, 10, 69, 626, 7812, 117650, 2097425, 43046803, 1000003158, 25937424602, 743008418676, 23298085122482, 793714774077816, 29192926025406980, 1152921504623628545, 48661191875666868482, 2185911559739084235093, 104127350297911241532842
Offset: 1

Views

Author

Seiichi Manyama, Jan 11 2023

Keywords

Crossrefs

Cf. A082245, A124923, A262843, A294956.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, #^(# + n/# - 2) &]; Array[a, 20] (* Amiram Eldar, Aug 14 2023 *)
  • PARI
    a(n) = sumdiv(n, d, d^(d+n/d-2));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, k^(k-1)*x^k/(1-k*x^k)))

Formula

G.f.: Sum_{k>0} k^(k-1) * x^k / (1 - k * x^k).
If p is prime, a(p) = 1 + p^(p-1).
Showing 1-6 of 6 results.

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

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