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 1-5 of 5 results.

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.

A308698 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} d^(k*d).

Original entry on oeis.org

1, 1, 2, 1, 5, 2, 1, 17, 28, 3, 1, 65, 730, 261, 2, 1, 257, 19684, 65553, 3126, 4, 1, 1025, 531442, 16777281, 9765626, 46688, 2, 1, 4097, 14348908, 4294967553, 30517578126, 2176783082, 823544, 4, 1, 16385, 387420490, 1099511628801, 95367431640626, 101559956688164, 678223072850, 16777477, 3
Offset: 1

Views

Author

Seiichi Manyama, Jun 17 2019

Keywords

Examples

			Square array begins:
   1,    1,       1,           1,              1, ...
   2,    5,      17,          65,            257, ...
   2,   28,     730,       19684,         531442, ...
   3,  261,   65553,    16777281,     4294967553, ...
   2, 3126, 9765626, 30517578126, 95367431640626, ...

Links

Crossrefs

Columns k=0..3 give A000005, A062796, A308696, A308697.
Row n=1..2 give A000012, A052539.
Cf. A308509, A308701, A308704.

Programs

  • Mathematica
    T[n_, k_] := DivisorSum[n, #^(k*#) &]; 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-1))).
G.f. of column k: Sum_{j>=1} j^(k*j) * x^j/(1 - x^j).

A308690 Square array A(n,k), n >= 1, k >= 0, where A(n,k) = Sum_{d|n} d^(k*n/d - k + 1), read by antidiagonals.

Original entry on oeis.org

1, 1, 3, 1, 3, 4, 1, 3, 4, 7, 1, 3, 4, 9, 6, 1, 3, 4, 13, 6, 12, 1, 3, 4, 21, 6, 24, 8, 1, 3, 4, 37, 6, 66, 8, 15, 1, 3, 4, 69, 6, 216, 8, 41, 13, 1, 3, 4, 133, 6, 762, 8, 201, 37, 18, 1, 3, 4, 261, 6, 2784, 8, 1289, 253, 68, 12, 1, 3, 4, 517, 6, 10386, 8, 9225, 2197, 648, 12, 28
Offset: 1

Views

Author

Seiichi Manyama, Jun 17 2019

Keywords

Examples

			Square array begins:
    1,  1,  1,   1,   1,    1,     1, ...
    3,  3,  3,   3,   3,    3,     3, ...
    4,  4,  4,   4,   4,    4,     4, ...
    7,  9, 13,  21,  37,   69,   133, ...
    6,  6,  6,   6,   6,    6,     6, ...
   12, 24, 66, 216, 762, 2784, 10386, ...
    8,  8,  8,   8,   8,    8,     8, ...

Links

Crossrefs

Columns k=0..3 give A000203, A055225, A308688, A308689.
Cf. A294579, A308509, A308694.

Programs

  • Mathematica
    T[n_, k_] := DivisorSum[n, #^(k*n/# - k + 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)).
A(p,k) = p+1 for prime p.

A308569 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} d^(k*n).

Original entry on oeis.org

1, 1, 2, 1, 5, 2, 1, 17, 28, 3, 1, 65, 730, 273, 2, 1, 257, 19684, 65793, 3126, 4, 1, 1025, 531442, 16781313, 9765626, 47450, 2, 1, 4097, 14348908, 4295032833, 30517578126, 2177317874, 823544, 4, 1, 16385, 387420490, 1099512676353, 95367431640626, 101560344351050, 678223072850, 16843009, 3
Offset: 1

Views

Author

Seiichi Manyama, Jun 08 2019

Keywords

Examples

			Square array begins:
   1,    1,       1,           1,              1, ...
   2,    5,      17,          65,            257, ...
   2,   28,     730,       19684,         531442, ...
   3,  273,   65793,    16781313,     4295032833, ...
   2, 3126, 9765626, 30517578126, 95367431640626, ...

Links

Crossrefs

Columns k=0..2 give A000005, A023887, A308570.
Rows n=1..2 give A000012, A052539.
A(n,n) gives A308571.
Cf. A308504, A308509.

Programs

  • Mathematica
    T[n_, k_] := DivisorSum[n, #^(k*n) &]; Table[T[k, n - k], {n, 1, 9}, {k, 1, n}] // Flatten (* Amiram Eldar, May 11 2021 *)
  • PARI
    T(n,k) = sumdiv(n, d, d^(k*n));
matrix(5, 5, n, k, T(n,k-1)) \\ Michel Marcus, Jun 08 2019

Formula

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

A308593 a(n) = Sum_{d|n} d^(n^2/d).

Original entry on oeis.org

1, 5, 28, 513, 3126, 840242, 823544, 8606711809, 7625984905477, 1221277338483250, 285311670612, 89215914432866222355906, 302875106592254, 316913110043605007120962336162, 608295209422788113565012727970423808, 680564733921105089459460296530789924865
Offset: 1

Views

Author

Seiichi Manyama, Jun 09 2019

Keywords

Links

Crossrefs

Diagonal of A308509.
Cf. A023887, A055225, A308571.

Programs

  • Mathematica
    Table[Sum[d^(n^2/d), {d, Divisors[n]}], {n,1,20}] (* Vaclav Kotesovec, Jun 09 2019 *)
  • PARI
    {a(n) = sumdiv(n, d, d^(n^2/d))}
Showing 1-5 of 5 results.

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