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 20 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.

A324159 Expansion of Sum_{k>=1} k * x^k / (1 - k * x^k)^k.

Original entry on oeis.org

1, 3, 4, 13, 6, 58, 8, 137, 172, 296, 12, 2063, 14, 1254, 5536, 7697, 18, 25201, 20, 68976, 70862, 23882, 24, 607485, 218776, 108720, 918568, 1810089, 30, 6746147, 32, 9408545, 11779582, 2233172, 19935756, 102405280, 38, 9968370, 145283360, 393585971, 42, 730233631, 44, 1296043651, 2718300016
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 02 2019

Keywords

Links

Crossrefs

Cf. A055225, A087909, A157019, A157020, A324158.

Programs

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

Formula

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

A327578 a(n) = n! * Sum_{d|n} d^(n/d - 1) / d!.

Original entry on oeis.org

1, 3, 7, 49, 121, 2521, 5041, 208321, 907201, 32810401, 39916801, 10621860481, 6227020801, 2877004690561, 19233710496001, 1415779600435201, 355687428096001, 1085522620595212801, 121645100408832001, 653741050484890368001, 6259137133527742464001, 576612373659657208473601
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 17 2019

Keywords

Links

Crossrefs

Cf. A057625, A087909, A327579.

Programs

  • Mathematica
    a[n_] := n! Sum[d^(n/d - 1)/d!, {d, Divisors[n]}]; Table[a[n], {n, 1, 22}]
nmax = 22; CoefficientList[Series[Sum[x^k/(k! (1 - k x^k)), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
  • PARI
    a(n) = n! * sumdiv(n, d, d^(n/d - 1) / d!); \\ Michel Marcus, Sep 17 2019

Formula

E.g.f.: Sum_{k>=1} x^k / (k! * (1 - k * x^k)).

A164941 a(n) = Sum_{d|n} phi(n/d)^(d-1).

Original entry on oeis.org

1, 2, 2, 3, 2, 5, 2, 5, 6, 7, 2, 17, 2, 9, 34, 15, 2, 45, 2, 87, 102, 13, 2, 191, 258, 15, 294, 289, 2, 1579, 2, 203, 1126, 19, 5394, 2577, 2, 21, 4242, 17227, 2, 16083, 2, 2037, 83282, 25, 2, 36107, 46658, 262423, 65794, 5839, 2, 139161, 1058578, 292455
Offset: 1

Views

Author

Franklin T. Adams-Watters, Sep 01 2009

Keywords

Links

Crossrefs

Cf. A000010, A087909, A309369.

Programs

  • PARI
    a(n) = sumdiv(n, d, eulerphi(d)^(n/d-1)); \\ Seiichi Manyama, Mar 13 2021
  • PARI
    a(n) = sum(k=1, n, eulerphi(n/gcd(k, n))^(gcd(k, n)-2)); \\ Seiichi Manyama, Mar 13 2021
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-eulerphi(k)*x^k))) \\ Seiichi Manyama, Mar 13 2021

Formula

G.f.: Sum_{k>=1} x^k/(1-phi(k)*x^k).
From Seiichi Manyama, Mar 13 2021: (Start)
a(n) = Sum_{k=1..n} phi(n/gcd(k, n))^(gcd(k, n) - 2).
If p is prime, a(p) = 2. (End)

A308694 Square array A(n,k), n >= 1, k >= 0, where A(n,k) = Sum_{d|n} d^(k*(n/d - 1)), read by antidiagonals.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 1, 2, 2, 3, 1, 2, 2, 4, 2, 1, 2, 2, 6, 2, 4, 1, 2, 2, 10, 2, 9, 2, 1, 2, 2, 18, 2, 27, 2, 4, 1, 2, 2, 34, 2, 93, 2, 14, 3, 1, 2, 2, 66, 2, 339, 2, 82, 11, 4, 1, 2, 2, 130, 2, 1269, 2, 578, 83, 23, 2, 1, 2, 2, 258, 2, 4827, 2, 4354, 731, 283, 2, 6
Offset: 1

Views

Author

Seiichi Manyama, Jun 17 2019

Keywords

Examples

			Square array begins:
   1, 1,  1,  1,   1,    1,    1, ...
   2, 2,  2,  2,   2,    2,    2, ...
   2, 2,  2,  2,   2,    2,    2, ...
   3, 4,  6, 10,  18,   34,   66, ...
   2, 2,  2,  2,   2,    2,    2, ...
   4, 9, 27, 93, 339, 1269, 4827, ...
   2, 2,  2,  2,   2,    2,    2, ...

Links

Crossrefs

Columns k=0..3 give A000005, A087909, A308692, A308693.
Cf. A308509, A308690.

Programs

  • Mathematica
    T[n_, k_] := DivisorSum[n, #^(k*(n/# - 1)) &]; Table[T[k, n - k], {n, 1, 12}, {k, 1, n}] // Flatten (* Amiram Eldar, May 09 2021 *)

Formula

L.g.f. of column k: -log(Product_{j>=1} (1 - j^k*x^j)^(1/j^(k+1))).
A(p,k) = 2 for prime p.

A308688 a(n) = Sum_{d|n} d^(2*n/d - 1).

Original entry on oeis.org

1, 3, 4, 13, 6, 66, 8, 201, 253, 648, 12, 5488, 14, 8550, 22824, 49681, 18, 316743, 20, 865578, 1611152, 2098506, 24, 27246276, 1953151, 33556656, 129199240, 202152908, 30, 1758141606, 32, 3223326753, 10460514288, 8589939540, 1261056768, 146050621105, 38
Offset: 1

Views

Author

Seiichi Manyama, Jun 17 2019

Keywords

Links

Crossrefs

Column k=2 of A308690.
Cf. A087909, A308692.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, #^(2*n/# - 1) &]; Array[a, 37] (* Amiram Eldar, May 09 2021 *)
  • PARI
    {a(n) = sumdiv(n, d, d^(2*n/d-1))}
  • PARI
    N=66; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, (1-k^2*x^k)^(1/k^2)))))

Formula

L.g.f.: -log(Product_{k>=1} (1 - k^2*x^k)^(1/k^2)) = Sum_{k>=1} a(k)*x^k/k.
a(p) = p+1 for prime p.
G.f.: Sum_{k>=1} k*x^k/(1 - k^2*x^k). - Ilya Gutkovskiy, Jul 25 2019

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

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.

A308367 Expansion of Sum_{k>=1} x^k/(1 + k*x^k).

Original entry on oeis.org

1, 0, 2, -2, 2, 1, 2, -12, 11, 11, 2, -49, 2, 57, 108, -200, 2, 40, 2, -391, 780, 1013, 2, -5423, 627, 4083, 6644, -4453, 2, -5043, 2, -49680, 59172, 65519, 18028, -251062, 2, 262125, 531612, -861481, 2, -515723, 2, -1049929, 5180382, 4194281, 2, -27246019, 117651
Offset: 1

Views

Author

Ilya Gutkovskiy, May 22 2019

Keywords

Links

Crossrefs

Cf. A076717, A087909, A300786.

Programs

  • Mathematica
    nmax = 49; CoefficientList[Series[Sum[x^k /(1 + k x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
nmax = 49; CoefficientList[Series[Log[Product[(1 + k x^k)^(1/k^2), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax] // Rest
Table[Sum[(-d)^(n/d - 1), {d, Divisors[n]}], {n, 1, 49}]
  • PARI
    a(n) = sumdiv(n, d, (-d)^(n/d-1)); \\ Michel Marcus, Mar 22 2021

Formula

L.g.f.: log(Product_{k>=1} (1 + k*x^k)^(1/k^2)) = Sum_{n>=1} a(n)*x^n/n.
a(n) = Sum_{d|n} (-d)^(n/d-1).
a(n) = 2 if n is odd prime.

A327249 Expansion of Sum_{k>=1} x^k * (1 + k * x^k)^k.

Original entry on oeis.org

1, 2, 1, 5, 1, 14, 1, 17, 28, 26, 1, 160, 1, 50, 251, 321, 1, 622, 1, 1607, 1030, 122, 1, 6257, 3126, 170, 2917, 12202, 1, 27291, 1, 28929, 6656, 290, 84036, 117721, 1, 362, 13183, 407121, 1, 417881, 1, 220100, 850312, 530, 1, 2246465, 823544, 2100626
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 15 2019

Keywords

Crossrefs

Cf. A006005 (positions of 1's), A087909, A217668, A260180, A327238.

Programs

  • Magma
    [&+[(n div d)^(d-1)*Binomial(n div d,d-1):d in Divisors(n)]:n in [1..50]]; // Marius A. Burtea, Sep 15 2019
  • Mathematica
    nmax = 50; CoefficientList[Series[Sum[x^k (1 + k x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, (n/#)^(# - 1) Binomial[n/#, # - 1] &], {n, 1, 50}]
  • PARI
    a(n) = sumdiv(n, d, (n/d)^(d-1) * binomial(n/d,d-1)); \\ Michel Marcus, Sep 15 2019

Formula

a(n) = Sum_{d|n} (n/d)^(d-1) * binomial(n/d,d-1).
Showing 1-10 of 20 results. Next

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