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 1-3 of 3 results.

A362683 Expansion of Sum_{k>0} (1/(1 - k*x^k)^2 - 1).

Original entry on oeis.org

2, 7, 10, 25, 16, 78, 22, 153, 136, 298, 34, 1254, 40, 1214, 2004, 3825, 52, 11385, 58, 20894, 18932, 25006, 70, 150002, 18826, 115274, 199828, 389510, 88, 1334624, 94, 1725281, 2131188, 2360266, 725948, 14878299, 112, 10486958, 22329428, 37317986, 124, 120957336, 130
Offset: 1

Views

Author

Seiichi Manyama, Jun 13 2023

Keywords

Crossrefs

Cf. A338662, A363639, A363640.
Cf. A007503, A055225, A359103, A363646.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^# * (# + 1) &]; Array[a, 50] (* Amiram Eldar, Jul 17 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^d*(d+1));

Formula

a(n) = Sum_{d|n} (n/d)^d * (d+1) = A055225(n) + A359103(n).
If p is prime, a(p) = 1 + 3*p.

A359882 a(n) = Sum_{d|n} d^n * (n/d)^d.

Original entry on oeis.org

1, 6, 30, 324, 3130, 53070, 823550, 17829896, 387951939, 10312525610, 285311670622, 9056807631948, 302875106592266, 11198819379685518, 437901307945957140, 18518802767263301648, 827240261886336764194, 39423330565860716459946
Offset: 1

Views

Author

Seiichi Manyama, Jan 16 2023

Keywords

Crossrefs

Cf. A066108, A308594, A359103.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, #^n * (n/#)^# &]; Array[a, 20] (* Amiram Eldar, Aug 09 2023 *)
  • PARI
    a(n) = sumdiv(n, d, d^n*(n/d)^d);

Formula

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

A359188 a(n) = Sum_{d|n} mu(n/d) * d * (n/d)^(d-1), where mu() is the Moebius function (A008683).

Original entry on oeis.org

1, 1, 2, 0, 4, -11, 6, -24, -18, -79, 10, -276, 12, -447, -464, -1008, 16, -3636, 18, -5580, -5228, -11263, 22, -41184, -3100, -53247, -59022, -116004, 28, -454501, 30, -524256, -649868, -1114111, -121344, -4438368, 36, -4980735, -6909200, -11106720, 40, -44114197, 42
Offset: 1

Views

Author

Seiichi Manyama, Dec 19 2022

Keywords

Crossrefs

Cf. A000010, A008683, A167531, A359103.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, MoebiusMu[n/#] * # * (n/#)^(#-1) &]; Array[a, 45] (* Amiram Eldar, Aug 27 2023 *)
  • PARI
    a(n) = sumdiv(n, d, moebius(n/d)*d*(n/d)^(d-1));
  • PARI
    my(N=50, x='x+O('x^N)); Vec(sum(k=1, N, moebius(k)*x^k/(1-k*x^k)^2))

Formula

G.f.: Sum_{k>=1} mu(k) * x^k/(1 - k * x^k)^2.
If p is prime, a(p) = p - 1.
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.