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 1-10 of 11 results. Next

A324158 Expansion of Sum_{k>=1} x^k / (1 - k * x^k)^k.

Original entry on oeis.org

1, 2, 2, 6, 2, 23, 2, 50, 56, 107, 2, 660, 2, 499, 1592, 2370, 2, 8246, 2, 18557, 21786, 11387, 2, 175198, 43752, 53419, 298892, 487762, 2, 1891098, 2, 2552066, 3905222, 1114403, 3785462, 29081597, 2, 4981099, 48376512, 95510772, 2, 218764940, 2, 346411232, 770590352
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 02 2019

Keywords

Links

Crossrefs

Cf. A055225, A087909, A157019, A157020, A167531, A324159.

Programs

  • Mathematica
    nmax = 45; CoefficientList[Series[Sum[x^k/(1 - k x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[Sum[(n/d)^(d - 1) Binomial[n/d + d - 2, d - 1], {d, Divisors[n]}], {n, 1, 45}]
  • PARI
    a(n) = sumdiv(n, d, (n/d)^(d-1) * binomial(n/d+d-2, d-1)); \\ Michel Marcus, Sep 02 2019
  • PARI
    N=66; x='x+O('x^N); Vec(sum(k=1, N, x^k/(1-k*x^k)^k)) \\ Seiichi Manyama, Sep 03 2019

Formula

a(n) = Sum_{d|n} (n/d)^(d-1) * binomial(n/d+d-2,d-1).
a(p) = 2, where p is prime.

A359112 a(n) = Sum_{d|n} (n/d) * d^(n-d).

Original entry on oeis.org

1, 3, 4, 13, 6, 109, 8, 777, 2197, 7541, 12, 374809, 14, 1675773, 31954096, 100794385, 18, 7391871271, 20, 163547770441, 2037381161992, 570634875581, 24, 1275177760626097, 476837158203151, 605750431288341, 450286447756825720, 2258377795760750777, 30
Offset: 1

Views

Author

Seiichi Manyama, Dec 17 2022

Keywords

Crossrefs

Cf. A000203, A167531.
Cf. A082245, A342628.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, #^(n-#)*n/# &]; Array[a, 29] (* Amiram Eldar, Aug 27 2023 *)
  • PARI
    a(n) = sumdiv(n, d, n/d*d^(n-d));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-(k*x)^k)^2))

Formula

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

A363642 Expansion of Sum_{k>0} x^k/(1 - k*x^k)^3.

Original entry on oeis.org

1, 4, 7, 17, 16, 55, 29, 129, 100, 311, 67, 1135, 92, 1919, 1486, 5409, 154, 17038, 191, 33491, 20938, 67871, 277, 262861, 9701, 373127, 296110, 978727, 436, 3134821, 497, 5051969, 3898522, 10027655, 474146, 39352069, 704, 49808159, 48362926, 127403221, 862, 411286429, 947
Offset: 1

Views

Author

Seiichi Manyama, Jun 13 2023

Keywords

Crossrefs

Cf. A363639, A363643, A363644.
Cf. A167531, A363645.
Cf. A007437, A363650.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^(#-1) * Binomial[# + 1, 2] &]; Array[a, 50] (* Amiram Eldar, Jul 18 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^(d-1)*binomial(d+1, 2));

Formula

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

A359103 a(n) = Sum_{d|n} d * (n/d)^d.

Original entry on oeis.org

1, 4, 6, 16, 10, 54, 14, 112, 99, 230, 22, 996, 26, 1022, 1620, 3232, 34, 9828, 38, 18100, 16380, 22814, 46, 133272, 15675, 106886, 179388, 354116, 58, 1218150, 62, 1589824, 1952676, 2228870, 630980, 13767264, 74, 9962270, 20732868, 34787000, 82, 113676402, 86
Offset: 1

Views

Author

Seiichi Manyama, Dec 17 2022

Keywords

Links

Crossrefs

Cf. A055225, A087909, A167531, A359112, A359863.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^#*# &]; Array[a, 43] (* Amiram Eldar, Aug 27 2023 *)
  • PARI
    a(n) = sumdiv(n, d, d*(n/d)^d);
  • PARI
    my(N=50, x='x+O('x^N)); Vec(sum(k=1, N, k*x^k/(1-k*x^k)^2))

Formula

a(n) = n * A087909(n).
G.f.: Sum_{k>=1} k * x^k/(1 - k * x^k)^2.
If p is prime, a(p) = 2 * p.
a(n) = [x^n] Sum_{k>0} k * (n * x / k)^k / (1 - x^k). - Seiichi Manyama, Jan 16 2023

A359018 a(n) = Sum_{d|n} d * 3^(d-1).

Original entry on oeis.org

1, 7, 28, 115, 406, 1492, 5104, 17611, 59077, 197242, 649540, 2127364, 6908734, 22325632, 71744968, 229600123, 731794258, 2324583475, 7360989292, 23245426690, 73222477552, 230128420012, 721764371008, 2259438436708, 7060738412431, 22029510754258, 68630377423960
Offset: 1

Views

Author

Seiichi Manyama, Dec 19 2022

Keywords

Links

Crossrefs

Cf. A002129, A034730, A083413, A167531, A359186, A359189.

Programs

  • Magma
    A359018:= func< n | (&+[3^(d-1)*d: d in Divisors(n)]) >;
[A359018(n): n in [1..40]]; // G. C. Greubel, Jun 26 2024
  • Mathematica
    a[n_] := DivisorSum[n, 3^(#-1)*# &]; Array[a, 27] (* Amiram Eldar, Aug 27 2023 *)
  • PARI
    a(n) = sumdiv(n, d, d*3^(d-1));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-3*x^k)^2))
  • SageMath
    def A359018(n): return sum(3^(k-1)*k for k in (1..n) if (k).divides(n))
[A359018(n) for n in range(1,41)] # G. C. Greubel, Jun 26 2024

Formula

G.f.: Sum_{k>=1} x^k/(1 - 3 * x^k)^2.

A363645 Expansion of Sum_{k>0} x^k/(1 - k*x^k)^4.

Original entry on oeis.org

1, 5, 11, 29, 36, 109, 85, 297, 256, 801, 287, 2881, 456, 5965, 3766, 17489, 970, 57385, 1331, 125681, 63498, 294933, 2301, 1072865, 24801, 1867009, 1087030, 4942561, 4496, 15697761, 5457, 28721057, 16895770, 63511593, 1404306, 225177013, 9140, 348661477
Offset: 1

Views

Author

Seiichi Manyama, Jun 13 2023

Keywords

Crossrefs

Cf. A167531, A363642.
Cf. A059358.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^(# - 1)*Binomial[# + 2, 3] &]; Array[a, 40] (* Amiram Eldar, Jul 18 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^(d-1)*binomial(d+2, 3));

Formula

a(n) = Sum_{d|n} (n/d)^(d-1) * binomial(d+2,3).

A363663 a(n) = Sum_{d|n} (n/d)^(d-1) * binomial(d+n-1,n).

Original entry on oeis.org

1, 4, 11, 46, 127, 596, 1717, 7792, 24806, 108450, 352717, 1563914, 5200301, 22539046, 77876117, 331982444, 1166803111, 4945693769, 17672631901, 74053888812, 269344740908, 1118110015874, 4116715363801, 16984153623296, 63205318063252, 259049084680612
Offset: 1

Views

Author

Seiichi Manyama, Jun 14 2023

Keywords

Crossrefs

Cf. A167531, A363642, A363645.
Cf. A343548, A363661, A363664.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^(#-1) * Binomial[# + n - 1, n] &]; Array[a, 25] (* Amiram Eldar, Jul 12 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^(d-1)*binomial(d+n-1, n));

Formula

a(n) = [x^n] Sum_{k>0} x^k/(1 - k*x^k)^(n+1).

A359186 a(n) = Sum_{d|n} d * 4^(d-1).

Original entry on oeis.org

1, 9, 49, 265, 1281, 6201, 28673, 131337, 589873, 2622729, 11534337, 50338105, 218103809, 939552777, 4026533169, 17180000521, 73014444033, 309238241337, 1305670057985, 5497560761865, 23089744212017, 96757034778633, 404620279021569, 1688849910733113
Offset: 1

Views

Author

Seiichi Manyama, Dec 19 2022

Keywords

Crossrefs

Cf. A002129, A083413, A359018.
Cf. A167531, A339684, A359190.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, 4^(#-1)*# &]; Array[a, 24] (* Amiram Eldar, Aug 27 2023 *)
  • PARI
    a(n) = sumdiv(n, d, d*4^(d-1));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-4*x^k)^2))

Formula

G.f.: Sum_{k>=1} x^k/(1 - 4 * x^k)^2.

A363641 Expansion of Sum_{k>0} x^(2*k)/(1 - k*x^k)^2.

Original entry on oeis.org

0, 1, 2, 4, 4, 10, 6, 20, 14, 42, 10, 127, 12, 206, 132, 512, 16, 1459, 18, 2655, 1492, 5142, 22, 17795, 524, 24602, 17540, 59567, 28, 177776, 30, 274656, 196884, 524322, 20156, 1901506, 36, 2359334, 2125828, 5682323, 40, 17453224, 42, 24641943, 22948512, 46137390, 46
Offset: 1

Views

Author

Seiichi Manyama, Jun 13 2023

Keywords

Crossrefs

Cf. A167531, A362683.
Cf. A363649.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^(#-2) * (#-1) &]; Array[a, 50] (* Amiram Eldar, Jul 18 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^(d-2)*(d-1));

Formula

a(n) = Sum_{d|n} (n/d)^(d-2) * (d-1).
If p is prime, a(p) = p - 1.

A363666 a(n) = Sum_{d|n} (n/d)^(d-1) * binomial(d+n-2,n-1).

Original entry on oeis.org

1, 3, 7, 29, 71, 355, 925, 4425, 13276, 60111, 184757, 856357, 2704157, 12137147, 40367461, 176999505, 601080391, 2616894901, 9075135301, 38884056181, 138014377810, 583674491643, 2104098963721, 8823912454489, 32247616479976, 133998376789707
Offset: 1

Views

Author

Seiichi Manyama, Jun 14 2023

Keywords

Crossrefs

Cf. A167531, A363642, A363645.
Cf. A332508, A363663, A363667.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^(#-1) * Binomial[# + n - 2, n - 1] &]; Array[a, 25] (* Amiram Eldar, Jul 12 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^(d-1)*binomial(d+n-2, n-1));

Formula

a(n) = [x^n] Sum_{k>0} x^k/(1 - k*x^k)^n.
Showing 1-10 of 11 results. Next

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