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.

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A332533 a(n) = (1/n) * Sum_{k=1..n} floor(n/k) * n^k.

Original entry on oeis.org

1, 4, 15, 92, 790, 9384, 137326, 2397352, 48428487, 1111122360, 28531183329, 810554859732, 25239592620853, 854769763924104, 31278135039463245, 1229782938533709200, 51702516368332126932, 2314494592676172411516, 109912203092257573556274, 5518821052632117898282620
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 16 2020

Keywords

Links

Crossrefs

Cf. A024916, A268235, A308313, A308814, A319194, A320095.
Sums of the form Sum_{k=1..n} q^(k-1)*floor(n/k): A344820 (q=-n), A344819 (q=-4), A344818 (q=-3), A344817 (q=-2), A059851 (q=-1), A006218 (q=1), A268235 (q=2), A344814 (q=3), A344815 (q=4), A344816 (q=5), this sequence (q=n).

Programs

  • Magma
    A332533:= func< n | (&+[Floor(n/j)*n^(j-1): j in [1..n]]) >;
[A332533(n): n in [1..40]]; // G. C. Greubel, Jun 27 2024
  • Maple
    seq(add(n^(k-1)*floor(n/k), k=1..n), n=1..60); # Ridouane Oudra, Mar 05 2023
  • Mathematica
    Table[(1/n) Sum[Floor[n/k] n^k, {k, 1, n}], {n, 1, 20}]
Table[(1/n) Sum[Sum[n^d, {d, Divisors[k]}], {k, 1, n}], {n, 1, 20}]
Table[SeriesCoefficient[(1/(1 - x)) Sum[x^k/(1 - n x^k), {k, 1, n}], {x, 0, n}], {n, 1, 20}]
  • PARI
    a(n) = sum(k=1, n, (n\k)*n^k)/n; \\ Michel Marcus, Feb 16 2020
  • PARI
    a(n) = sum(k=1, n, sumdiv(k, d, n^(d-1))); \\ Seiichi Manyama, May 29 2021
  • SageMath
    def A332533(n): return sum((n//j)*n^(j-1) for j in range(1,n+1))
[A332533(n) for n in range(1,41)] # G. C. Greubel, Jun 27 2024

Formula

a(n) = [x^n] (1/(1 - x)) * Sum_{k>=1} x^k / (1 - n*x^k).
a(n) = (1/n) * Sum_{k=1..n} Sum_{d|k} n^d.
a(n) ~ n^(n-1). - Vaclav Kotesovec, May 28 2021
a(n) = (1/(n-1)) * Sum_{k=1..n} (n^floor(n/k) - 1), for n>=2. - Ridouane Oudra, Mar 05 2023

A366919 a(n) = Sum_{k=1..n} (-1)^k*k^n*floor(n/k).

Original entry on oeis.org

-1, 2, -22, 203, -2285, 33855, -609345, 12420372, -284964519, 7347342215, -209807114169, 6554034238459, -222469737401739, 8159109186320903, -321461264348047819, 13538455640979049698, -606976994365011212414, 28864017965496692865925, -1451086990386146504580735
Offset: 1

Views

Author

Chai Wah Wu, Oct 28 2023

Keywords

Crossrefs

Cf. A024919, A308313, A366915, A366917, A319649.

Programs

  • Mathematica
    a[n_]:=Sum[ (-1)^k*k^n*Floor[n/k],{k,n}]; Array[a,19] (* Stefano Spezia, Oct 29 2023 *)
  • PARI
    a(n) = sum(k=1, n, (-1)^k*k^n*(n\k)); \\ Michel Marcus, Oct 29 2023
  • Python
    from math import isqrt
from sympy import bernoulli
def A366919(n): return ((((s:=isqrt(m:=n>>1))+1)*(bernoulli(n+1)-bernoulli(n+1,s+1))<

Formula

a(n) = (-1)^n*A308313(n).
Let A(n,k) = Sum_{j=1..n} j^k * floor(n/j). Then a(n) = 2^(n+1)*A(floor(n/2),n)-A(n,n).
Showing 1-2 of 2 results.

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