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

Author

Vladeta Jovovic, Jun 12 2004

Keywords

• easy
• nonn
• changed

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)

Crossrefs

Cf. A091626, A038548.

Cf. A006218, A000196, A211264.

Programs

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 *)
(* Clark Kimberling, Jun 18 2011 *)
Table[Sum[Boole[d <= Sqrt[n]], {d, Divisors[n]}], {n, 1, 66}] // Accumulate (* Jean-François Alcover, Dec 13 2012 *)

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

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

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

a(n) = A211264(n) + A000196(n). - Ridouane Oudra, Sep 13 2024