A094820 Partial sums of A038548.

1, 2, 3, 5, 6, 8, 9, 11, 13, 15, 16, 19, 20, 22, 24, 27, 28, 31, 32, 35, 37, 39, 40, 44, 46, 48, 50, 53, 54, 58, 59, 62, 64, 66, 68, 73, 74, 76, 78, 82, 83, 87, 88, 91, 94, 96, 97, 102, 104, 107, 109, 112, 113, 117, 119, 123, 125, 127, 128, 134, 135, 137, 140, 144, 146, 150

Offset

1,2

Comments

a(n) = number of pairs (c,d) of integers such that 0 < c <= d and c*d <= n. - Clark Kimberling, Jun 18 2011

Equivalently, the number of representations of n in the form x + y*z, where x, y, and z are positive integers and y <= z. - John W. Layman, Feb 21 2012

Links

Peter Kagey, Table of n, a(n) for n = 1..10000

Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Formula

G.f.: (1/(1 - x))*Sum_{k>=1} x^(k^2)/(1 - x^k). - Ilya Gutkovskiy, Apr 13 2017

a(n) ~ (log(n) + 2*gamma - 1)*n/2 + sqrt(n)/2, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Aug 19 2019

a(n) = (A006218(n) + A000196(n))/2. - Ridouane Oudra, Nov 25 2022

Maple

ListTools:-PartialSums([seq(ceil(numtheory:-tau(n)/2), n=1..100)]); # Robert Israel, Feb 24 2016

Mathematica

f[n_, k_] := Floor[n/k] - Floor[(n - 1)/k]
g[n_, k_] := If[k^2 <= n, f[n, k], 0]
Table[Sum[f[n, k], {k, 1, n}], {n, 1, 100}] (* A000005 *)
t = Table[Sum[g[n, k], {k, 1, n}], {n, 1, 100}]
(* A038548 *)
a[n_] := Sum[t[[i]], {i, 1, n}]
Table[a[n], {n, 1, 100}]  (* A094820 *)
(* from Clark Kimberling, Jun 18 2011 *)
Table[Sum[Boole[d <= Sqrt[n]], {d, Divisors[n]}], {n, 1, 66}] // Accumulate (* Jean-François Alcover, Dec 13 2012 *)

Prog

(Ruby)
  def a(n)
    (1..Math.sqrt(n)).inject(0) { |accum, i| accum + 1 + (n/i).to_i - i }
  end # Peter Kagey, Feb 24 2016
(PARI) a(n) = sum(k=1, n, ceil(numdiv(k)/2)); \\ Michel Marcus, Feb 24 2016
(Python)
from math import isqrt
def A094820(n): return ((s:=isqrt(n))*(1-s)>>1)+sum(n//k for k in range(1, s+1)) # Chai Wah Wu, Oct 23 2023

Crossrefs

Cf. A091626, A038548.

Cf. A006218, A000196.

Sequence in context: A050172 A191877 A277124 * A309793 A093689 A097702

Adjacent sequences: A094817 A094818 A094819 * A094821 A094822 A094823

Keyword

easy,nonn

Author

Vladeta Jovovic, Jun 12 2004

Status

approved