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 1-4 of 4 results.

A338693 a(n) = Sum_{d|n} d^(d - n/d) * binomial(d, n/d).

Original entry on oeis.org

1, 4, 27, 257, 3125, 46665, 823543, 16777312, 387420490, 10000001250, 285311670611, 8916100467712, 302875106592253, 11112006825910963, 437893890380859625, 18446744073716891649, 827240261886336764177, 39346408075296709766628, 1978419655660313589123979
Offset: 1

Views

Author

Seiichi Manyama, Apr 24 2021

Keywords

Links

Crossrefs

Cf. A318636, A318637, A318638, A338685, A338694.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, #^(# - n/#) * Binomial[#, n/#] &]; Array[a, 20] (* Amiram Eldar, Apr 24 2021 *)
  • PARI
    a(n) = sumdiv(n, d, d^(d-n/d)* binomial(d, n/d));
  • PARI
    N=20; x='x+O('x^N); Vec(sum(k=1, N, (k+x^k)^k-k^k))

Formula

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

A376014 a(n) = Sum_{d|n} d^d * binomial(n/d,d).

Original entry on oeis.org

1, 2, 3, 8, 5, 18, 7, 32, 36, 50, 11, 180, 13, 98, 285, 384, 17, 702, 19, 1480, 966, 242, 23, 5640, 3150, 338, 2295, 9352, 29, 22440, 31, 18432, 4488, 578, 65660, 85500, 37, 722, 7761, 229560, 41, 337302, 43, 85448, 406080, 1058, 47, 1449360, 823592, 788750, 18411
Offset: 1

Views

Author

Seiichi Manyama, Sep 06 2024

Keywords

Links

Crossrefs

Cf. A062796, A338694, A376018.

Programs

  • PARI
    a(n) = sumdiv(n, d, d^d*binomial(n/d, d));
  • PARI
    my(N=60, x='x+O('x^N)); Vec(sum(k=1, N, (k*x^k)^k/(1-x^k)^(k+1)))
  • Python
    from math import comb
from itertools import takewhile
from sympy import divisors
def A376014(n): return sum(d**d*comb(n//d,d) for d in takewhile(lambda d:d**2<=n,divisors(n))) # Chai Wah Wu, Sep 06 2024

Formula

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

A360759 a(n) = Sum_{d|n} d^(d+n/d) * binomial(d,n/d).

Original entry on oeis.org

1, 16, 243, 4112, 78125, 1680345, 40353607, 1073766400, 31381060338, 1000000781250, 34522712143931, 1283918489808640, 51185893014090757, 2177953338656796883, 98526125335697265625, 4722366482899710050304, 239072435685151324847153
Offset: 1

Views

Author

Seiichi Manyama, Feb 19 2023

Keywords

Links

Crossrefs

Cf. A318636, A327238, A338685, A338693, A338694.
Cf. A360712.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, #^(# + n/#) * Binomial[#, n/#] &]; Array[a, 20] (* Amiram Eldar, Aug 02 2023 *)
  • PARI
    a(n) = sumdiv(n, d, d^(d+n/d)*binomial(d, n/d));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, k^k*((1+k*x^k)^k-1)))

Formula

G.f.: Sum_{k>0} k^k * ( (1 + k*x^k)^k - 1 ).
If p is prime, a(p) = p^(p+2).

A338695 a(n) = Sum_{d|n} 2^(d-1) * binomial(d, n/d).

Original entry on oeis.org

1, 4, 12, 34, 80, 204, 448, 1072, 2308, 5280, 11264, 25088, 53248, 116032, 245920, 527880, 1114112, 2369152, 4980736, 10508880, 22022336, 46193664, 96468992, 201469408, 419430416, 872734720, 1811960832, 3758844096, 7784628224, 16107909312, 33285996544, 68723417856, 141734089728
Offset: 1

Views

Author

Seiichi Manyama, Apr 24 2021

Keywords

Crossrefs

Cf. A034729, A318636, A318637, A338694.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, 2^(# - 1) * Binomial[#, n/#] &]; Array[a, 20] (* Amiram Eldar, Apr 24 2021 *)
  • PARI
    a(n) = sumdiv(n, d, 2^(d-1)*binomial(d, n/d));
  • PARI
    N=40; x='x+O('x^N); Vec(sum(k=1, N, (2+2*x^k)^k-2^k)/2)

Formula

G.f.: (1/2) * Sum_{k>=1} ( (2 + 2 * x^k)^k - 2^k ) = Sum_{k>=1} 2^(k-1) * ( (1 + x^k)^k - 1 ).
If p is prime, a(p) = p * 2^(p-1).
Showing 1-4 of 4 results.

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