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.

A283534 Expansion of exp( Sum_{n>=1} -A283533(n)/n*x^n ) in powers of x.

Original entry on oeis.org

1, -1, -16, -713, -64687, -9688545, -2165715003, -675843665621, -280752874575386, -149800127959983890, -99844730502381895830, -81300082280849836639246, -79413710313923588156379547, -91652445699847071535357000689, -123383623610527054787988720527285, -191626051373071219208574650313032502
Offset: 0

Views

Author

Seiichi Manyama, Mar 10 2017

Keywords

Links

Crossrefs

Cf. Product_{k>=1} (1 - x^k)^(k^(m*k)): A010815 (m=0), A283499 (m=1), this sequence (m=2), A283536 (m=3).
Cf. A283579 (Product_{k>=1} 1/(1 - x^k)^(k^(2*k))).

Programs

  • Mathematica
    A[n_] :=  Sum[d^(2*d + 1), {d, Divisors[n]}]; a[n_] := If[n==0, 1, -(1/n)*Sum[A[k]*a[n - k], {k, n}]]; Table[a[n], {n, 0, 13}] (* Indranil Ghosh, Mar 11 2017 *)
  • PARI
    a(n) = if(n==0, 1, -(1/n)*sum(k=1, n, sumdiv(k, d, d^(2*d + 1))*a(n - k)));
for(n=0, 15, print1(a(n), ", ")) \\ Indranil Ghosh, Mar 11 2017

Formula

G.f.: Product_{k>=1} (1 - x^k)^(k^(2*k)).
a(n) = -(1/n)*Sum_{k=1..n} A283533(k)*a(n-k) for n > 0.

A283579 Expansion of exp( Sum_{n>=1} A283533(n)/n*x^n ) in powers of x.

Original entry on oeis.org

1, 1, 17, 746, 66418, 9843707, 2187941520, 680615139257, 282199700198462, 150389915598653924, 100155578743010743914, 81505577512720707466924, 79580089689432499741178617, 91814299713761739807846854872
Offset: 0

Views

Author

Seiichi Manyama, Mar 11 2017

Keywords

Links

Crossrefs

Cf. Product_{k>=1} 1/(1 - x^k)^(k^(m*k)): A000041 (m=0), A023880 (m=1), this sequence (m=2), A283580 (m=3).
Cf. A283534 (Product_{k>=1} (1 - x^k)^(k^(2*k))).

Programs

  • Mathematica
    A[n_] :=  Sum[d^(2*d + 1), {d, Divisors[n]}]; a[n_] := If[n==0, 1, (1/n)*Sum[A[k]*a[n - k], {k, n}]]; Table[a[n], {n, 0, 13}] (* Indranil Ghosh, Mar 11 2017 *)
  • PARI
    A(n) = sumdiv(n, d, d^(2*d + 1));
a(n) = if(n==0, 1, (1/n)*sum(k=1, n, A(k)*a(n - k)));
for(n=0, 11, print1(a(n),", ")) \\ Indranil Ghosh, Mar 11 2017

Formula

G.f.: Product_{k>=1} 1/(1 - x^k)^(k^(2*k)).
a(n) = (1/n)*Sum_{k=1..n} A283533(k)*a(n-k) for n > 0.
a(n) ~ n^(2*n) * (1 + exp(-2)/n^2). - Vaclav Kotesovec, Mar 17 2017

A283535 a(n) = Sum_{d|n} d^(3*d + 1).

Original entry on oeis.org

1, 129, 59050, 67108993, 152587890626, 609359740069674, 3909821048582988050, 37778931862957228818561, 523347633027360537213570571, 10000000000000000000152587890754, 255476698618765889551019445759400442, 8505622499821102144576132293474637113130
Offset: 1

Views

Author

Seiichi Manyama, Mar 10 2017

Keywords

Examples

			a(6) = 1^(3+1) + 2^(6+1) + 3^(9+1) + 6^(18+1) = 609359740069674.

Links

Crossrefs

Cf. Sum_{d|n} d^(k*d+1): A283498 (k=1), A283533 (k=2), this sequence (k=3).
Cf. A308697.

Programs

  • Mathematica
    f[n_] := Block[{d = Divisors[n]}, Total[d^(3 d + 1)]]; Array[f, 12] (* Robert G. Wilson v, Mar 10 2017 *)
  • PARI
    a(n) = sumdiv(n, d, d^(3*d+1)); \\ Michel Marcus, Mar 11 2017
  • PARI
    N=20; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, (1-x^k)^k^(3*k))))) \\ Seiichi Manyama, Jun 18 2019

Formula

L.g.f.: -log(Product_{k>=1} (1 - x^k)^(k^(3*k))) = Sum_{k>=1} a(k)*x^k/k. - Seiichi Manyama, Jun 18 2019

A308753 a(n) = Sum_{d|n} d^(2*(d-1)).

Original entry on oeis.org

1, 5, 82, 4101, 390626, 60466262, 13841287202, 4398046515205, 1853020188851923, 1000000000000390630, 672749994932560009202, 552061438912436478063702, 542800770374370512771595362, 629983141281877223617054459942
Offset: 1

Views

Author

Seiichi Manyama, Jun 22 2019

Keywords

Links

Crossrefs

Column k=2 of A308701.
Cf. A283533, A308692, A308696, A308755, A308756.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, #^(2*(# - 1)) &]; Array[a, 14] (* Amiram Eldar, May 08 2021 *)
  • PARI
    {a(n) = sumdiv(n, d, d^(2*(d-1)))}
  • PARI
    N=20; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, (1-x^k)^k^(2*k-3)))))
  • PARI
    N=20; x='x+O('x^N); Vec(sum(k=1, N, k^(2*(k-1))*x^k/(1-x^k)))

Formula

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

A283369 a(n) = Sum_{d|n} d^(4*d + 1).

Original entry on oeis.org

1, 513, 1594324, 17179869697, 476837158203126, 28430288029931296212, 3219905755813179726837608, 633825300114114700765531472385, 202755595904452569706561330874548093, 100000000000000000000000000476837158203638
Offset: 1

Views

Author

Seiichi Manyama, Mar 17 2017

Keywords

Examples

			a(6) = 1^(4+1) + 2^(8+1) + 3^(12+1) + 6^(24+1) = 28430288029931296212.

Links

Crossrefs

Cf. Sum_{d|n} d^(k*d+1): A283498 (k=1), A283533 (k=2), A283535 (k=3), this sequence (k=4).

Programs

  • Mathematica
    Table[Sum[d^(4*d + 1), {d, Divisors[n]}], {n, 20}] (* Indranil Ghosh, Mar 17 2017 *)
  • PARI
    for(n=1, 20, print1(sumdiv(n, d, d^(4*d + 1)),", ")) \\ Indranil Ghosh, Mar 17 2017

A308704 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} d^(k*d+1).

Original entry on oeis.org

1, 1, 3, 1, 9, 4, 1, 33, 82, 7, 1, 129, 2188, 1033, 6, 1, 513, 59050, 262177, 15626, 12, 1, 2049, 1594324, 67108993, 48828126, 280026, 8, 1, 8193, 43046722, 17179869697, 152587890626, 13060696236, 5764802, 15, 1, 32769, 1162261468, 4398046513153, 476837158203126, 609359740069674, 4747561509944, 134218761, 13
Offset: 1

Views

Author

Seiichi Manyama, Jun 18 2019

Keywords

Examples

			Square array begins:
   1,     1,        1,            1,               1, ...
   3,     9,       33,          129,             513, ...
   4,    82,     2188,        59050,         1594324, ...
   7,  1033,   262177,     67108993,     17179869697, ...
   6, 15626, 48828126, 152587890626, 476837158203126, ...

Links

Crossrefs

Columns k=0..3 give A000203, A283498, A283533, A283535.
Row n=1..2 give A000012, A087289.
Cf. A294579, A308698, A308701.

Programs

  • Mathematica
    T[n_, k_] := DivisorSum[n, #^(k*# + 1) &]; Table[T[k, n - k], {n, 1, 9}, {k, 1, n}] // Flatten (* Amiram Eldar, May 09 2021 *)

Formula

L.g.f. of column k: -log(Product_{j>=1} (1 - x^j)^(j^(k*j))).
G.f. of column k: Sum_{j>=1} j^(k*j+1) * x^j/(1 - x^j).

A308756 a(n) = Sum_{d|n} d^(2*(d-2)).

Original entry on oeis.org

1, 2, 10, 258, 15626, 1679627, 282475250, 68719476994, 22876792454971, 10000000000015627, 5559917313492231482, 3833759992447476802059, 3211838877954855105157370, 3214199700417740937033562867, 3787675244106352329254150406260
Offset: 1

Views

Author

Seiichi Manyama, Jun 22 2019

Keywords

Links

Crossrefs

Cf. A283533, A308696.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, #^(2*(# - 2)) &]; Array[a, 15] (* Amiram Eldar, May 08 2021 *)
  • PARI
    {a(n) = sumdiv(n, d, d^(2*(d-2)))}
  • PARI
    N=20; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, (1-x^k)^k^(2*k-5)))))
  • PARI
    N=20; x='x+O('x^N); Vec(sum(k=1, N, k^(2*(k-2))*x^k/(1-x^k)))

Formula

L.g.f.: -log(Product_{k>=1} (1 - x^k)^(k^(2*k-5))) = Sum_{k>=1} a(k)*x^k/k.
G.f.: Sum_{k>=1} k^(2*(k-2)) * x^k/(1 - x^k).
Showing 1-7 of 7 results.

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