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 1-4 of 4 results.

A308509 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} d^(k*n/d).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 5, 4, 3, 1, 9, 10, 9, 2, 1, 17, 28, 33, 6, 4, 1, 33, 82, 129, 26, 24, 2, 1, 65, 244, 513, 126, 182, 8, 4, 1, 129, 730, 2049, 626, 1458, 50, 41, 3, 1, 257, 2188, 8193, 3126, 11954, 344, 577, 37, 4, 1, 513, 6562, 32769, 15626, 99594, 2402, 8705, 811, 68, 2
Offset: 1

Views

Author

Seiichi Manyama, Jun 02 2019

Keywords

Examples

			Square array begins:
   1,  1,   1,    1,     1,     1,      1, ...
   2,  3,   5,    9,    17,    33,     65, ...
   2,  4,  10,   28,    82,   244,    730, ...
   3,  9,  33,  129,   513,  2049,   8193, ...
   2,  6,  26,  126,   626,  3126,  15626, ...
   4, 24, 182, 1458, 11954, 99594, 840242, ...

Links

Crossrefs

Columns k=0..3 give A000005, A055225, A073705, A073706.
Cf. A294579.

Programs

  • Mathematica
    T[n_, k_] := DivisorSum[n, #^(k*n/#) &]; Table[T[k, n - k], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, May 11 2021 *)
  • PARI
    T(n,k) = sumdiv(n, d, (n/d)^(k*d));
matrix(9, 9, n, k, T(n,k-1)) \\ Michel Marcus, Jun 02 2019

Formula

L.g.f. of column k: -log(Product_{j>=1} (1 - j^k*x^j)^(1/j)).
A(n,k) = Sum_{d|n} (n/d)^(k*d).

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.

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).

A294567 a(n) = Sum_{d|n} d^(1 + 2*n/d).

Original entry on oeis.org

1, 9, 28, 97, 126, 588, 344, 2049, 2917, 6174, 1332, 53764, 2198, 52320, 258648, 430081, 4914, 2463429, 6860, 8352582, 15181712, 8560308, 12168, 242240964, 48843751, 134606598, 1167064120, 1651526120, 24390, 14202123408, 29792, 25905102849, 94162701936
Offset: 1

Views

Author

Seiichi Manyama, Nov 02 2017

Keywords

Comments

If p is prime, a(p) = 1 + p^3. - Robert Israel, Nov 03 2017

Links

Crossrefs

Column k=2 of A294579.
Cf. A292164.

Programs

  • Maple
    f:= n -> add(d^(1+2*n/d),d=numtheory:-divisors(n)):
map(f, [$1..100]); # Robert Israel, Nov 03 2017
  • Mathematica
    sd[n_] := Module[{d = Divisors[n]}, Total[d^(1 + (2 n)/d)]]; Array[sd,40] (* Harvey P. Dale, Mar 17 2020 *)
a[n_] := DivisorSum[n, #^(1 + 2*n/#) &]; Array[a, 33] (* Amiram Eldar, Oct 04 2023 *)
  • PARI
    a(n) = sumdiv(n, d, d^(1+2*n/d)); \\ Michel Marcus, Nov 02 2017
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, k^3*x^k/(1-k^2*x^k))) \\ Seiichi Manyama, Jan 14 2023

Formula

L.g.f.: -log(Product_{k>=1} (1 - k^2*x^k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Mar 12 2018
G.f.: Sum_{k>0} k^3 * x^k / (1 - k^2 * x^k). - Seiichi Manyama, Jan 14 2023
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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