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 1-7 of 7 results.

A342628 a(n) = Sum_{d|n} d^(n-d).

Original entry on oeis.org

1, 2, 2, 6, 2, 45, 2, 322, 731, 3383, 2, 132901, 2, 827641, 10297068, 33570818, 2, 2578617270, 2, 44812807567, 678610493340, 285312719189, 2, 393061010002613, 95367431640627, 302875123369471, 150094917726535604, 569939345952661545, 2, 105474306078445349841, 2
Offset: 1

Views

Author

Seiichi Manyama, Mar 16 2021

Keywords

Links

Crossrefs

Cf. A023887, A062796, A308594, A342629.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, #^(n - #) &]; Array[a, 30] (* Amiram Eldar, Mar 17 2021 *)
  • PARI
    a(n) = sumdiv(n, d, d^(n-d));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-(k*x)^k)))
  • Python
    from sympy import divisors
def A342628(n): return sum(d**(n-d) for d in divisors(n,generator=True)) # Chai Wah Wu, Jun 19 2022

Formula

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

A342612 a(n) = Sum_{d|n} phi(n/d)^(n-d).

Original entry on oeis.org

1, 2, 5, 10, 257, 50, 46657, 16450, 1679681, 327682, 10000000001, 4196098, 8916100448257, 15237476354, 4398063289345, 35184640528386, 18446744073709551617, 19747769389058, 39346408075296537575425
Offset: 1

Views

Author

Seiichi Manyama, Mar 16 2021

Keywords

Links

Crossrefs

Cf. A000010, A342489, A342619, A342629.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, EulerPhi[n/#]^(n - #) &]; Array[a, 20] (* Amiram Eldar, Mar 17 2021 *)
  • PARI
    a(n) = sumdiv(n, d, eulerphi(n/d)^(n-d));
  • PARI
    a(n) = sum(k=1, n, eulerphi(n/gcd(k, n))^(n-gcd(k, n)-1));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)^(k-1)*x^k/(1-eulerphi(k)^(k-1)*x^k)))

Formula

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

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

Links

Crossrefs

Cf. A294645, A342629, A342675.

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.

A357051 a(n) = Sum_{d|n} 3^(n-d).

Original entry on oeis.org

1, 4, 10, 37, 82, 352, 730, 2998, 7291, 26488, 59050, 263170, 531442, 2127952, 5373460, 19669879, 43046722, 187086916, 387420490, 1607136634, 3878987860, 13947314752, 31381059610, 139902374692, 285916320883, 1129719740248, 2824682785300, 10460357985970
Offset: 1

Views

Author

Seiichi Manyama, Dec 20 2022

Keywords

Crossrefs

Cf. A074854, A112329, A359206.
Cf. A342628, A342629, A359203.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, 3^(n-#) &]; Array[a, 28] (* Amiram Eldar, Aug 23 2023 *)
  • PARI
    a(n) = sumdiv(n, d, 3^(n-d));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, 3^(k-1)*x^k/(1-3^(k-1)*x^k)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-(3*x)^k)))

Formula

G.f.: Sum_{k>=1} 3^(k-1) * x^k/(1 - 3^(k-1) * x^k).
G.f.: Sum_{k>=1} x^k/(1 - (3 * x)^k).

A358660 a(n) = Sum_{d|n} d * (n/d)^(n-d).

Original entry on oeis.org

1, 4, 12, 76, 630, 7968, 117656, 2105416, 43048917, 1000781420, 25937424612, 743130116112, 23298085122494, 793742455829456, 29192926758107760, 1152930300766980112, 48661191875666868498, 2185915267189632382650, 104127350297911241532860
Offset: 1

Views

Author

Seiichi Manyama, Dec 17 2022

Keywords

Links

Crossrefs

Cf. A090879, A342629, A356539, A359112.

Programs

  • Mathematica
    a[n_] := Total[Map[#*(n/#)^(n - #) &, Divisors[n]]];
Table[a[n],{n,1,100}]
a[n_] := DivisorSum[n, (n/#)^(n-#)*# &]; Array[a, 19] (* Amiram Eldar, Aug 27 2023 *)
  • PARI
    a(n) = sumdiv(n, d, d*(n/d)^(n-d));
  • 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)^2))

Formula

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

A359206 a(n) = Sum_{d|n} 4^(n-d).

Original entry on oeis.org

1, 5, 17, 81, 257, 1345, 4097, 20737, 69633, 328705, 1048577, 5574657, 16777217, 83902465, 286261249, 1359020033, 4294967297, 22565617665, 68719476737, 348967141377, 1168499539969, 5497562333185, 17592186044417, 93531519582209, 282574488338433
Offset: 1

Views

Author

Seiichi Manyama, Dec 20 2022

Keywords

Crossrefs

Cf. A074854, A112329, A357051.
Cf. A342628, A342629, A359204.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, 4^(n-#) &]; Array[a, 25] (* Amiram Eldar, Aug 23 2023 *)
  • PARI
    a(n) = sumdiv(n, d, 4^(n-d));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, 4^(k-1)*x^k/(1-4^(k-1)*x^k)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-(4*x)^k)))

Formula

G.f.: Sum_{k>=1} 4^(k-1) * x^k/(1 - 4^(k-1) * x^k).
G.f.: Sum_{k>=1} x^k/(1 - (4 * x)^k).

A359442 a(n) = Sum_{d|n} d^(n + 1 - d - n/d).

Original entry on oeis.org

1, 2, 2, 4, 2, 15, 2, 74, 83, 643, 2, 12635, 2, 117715, 397188, 2359426, 2, 103572204, 2, 1260918355, 13841818644, 25937425627, 2, 5612318393211, 152587890627, 23298085126579, 1853020231898564, 2422197090649523, 2, 1032944452284531101, 2, 10376297939508166658
Offset: 1

Views

Author

Seiichi Manyama, Jan 14 2023

Keywords

Links

Crossrefs

Cf. A294645, A342628, A342629, A342677, A359700.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, #^(n + 1 - # - n/#) &]; Array[a, 32] (* Amiram Eldar, Aug 09 2023 *)
  • PARI
    a(n) = sumdiv(n, d, d^(n+1-d-n/d));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-(k*x)^k/k)))

Formula

G.f.: Sum_{k>0} x^k / (1 - (k * x)^k / k).
If p is prime, a(p) = 2.
Showing 1-7 of 7 results.

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